This is (mainly) a survey of recent results on the problem
of the existence of infinitely many periodic orbits for Hamiltonian
diffeomorphisms and Reeb flows. We focus on the Conley conjecture,
proved for a broad class of closed symplectic manifolds, asserting
that under some natural conditions on the manifold every Hamiltonian
diffeomorphism has infinitely many (simple) periodic orbits. We
discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb
flows, the cases where the conjecture is known to fail, the question
of the generic existence of infinitely many periodic orbits, and
local geometrical conditions that force the existence of infinitely
many periodic orbits. We also show how a recently established
variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a
low-energy charge in a non-vanishing magnetic field on a surface
other than a sphere.

Hamiltonian systems tend to have infinitely many periodic orbits.
For many phase spaces, every system, without any restrictions, has
infinitely many simple periodic orbits. Moreover, even if not
holding unconditionally, this is still a $C^{\infty}$-generic property
of Hamiltonian systems for the majority of phase spaces. Finally,
for some phase spaces, a system has infinitely many simple periodic
orbits when certain natural local conditions are met.

This paper is mainly a survey focusing on this phenomenon for
Hamiltonian diffeomorphisms and Reeb flows. The central theme of the
paper is the so-called Conley conjecture, proved for a broad class
of closed symplectic manifolds and asserting that under some natural
conditions on the manifold every Hamiltonian diffeomorphism has
infinitely many (simple) periodic orbits. We discuss in detail the
established cases of the conjecture and related results, including
an analog of the conjecture for Reeb flows, and also the manifolds
for which the conjecture is known to fail. In particular, we
investigate local geometrical conditions that force globally the
existence of infinitely many periodic orbits and consider the
question of the generic existence of infinitely many periodic
orbits.

We also briefly touch upon the applications to dynamical systems of
physical origin. For instance, we show how a recently established
variant of the Conley conjecture for Reeb flows can be used to prove
the existence of infinitely many simple periodic orbits of a
low-energy charge in a non-vanishing magnetic field on a surface
other than a sphere.

Our perspective on the problem and the methods we use here are
mainly Morse theoretic, broadly understood, and homological. In this
framework, the reasons for the existence of periodic orbits lie at
the interplay between local dynamical features of the system and the
global behavior of the homology “counting” the periodic orbits,
e.g., Floer, symplectic or contact homology.

This is not the only perspective on the subject. For instance, in
dimensions two and three, one can alternatively use exceptionally
powerful methods of low-dimensional dynamics (see, e.g., [Franks1992]; [Franks1996]; [Franks and Handel2003]; [Le Calvez2006]) and holomorphic curves (see, e.g., [Bramham and Hofer2012]; [Hofer et al.1998]; [Hofer et al.2003]). In higher dimensions, however, most of the
results on this class of problems rely on homological methods.

It is important to note that for Hamiltonian diffeomorphisms, in
contrast with the classical setting of geodesic flows on a majority
of manifolds as in [Gromoll and Meyer1969], the existence of infinitely many
simple periodic orbits is not forced by the homological growth.
Likewise, the local dynamical features we consider here are usually
of different flavor from, say, homoclinic intersections or elliptic
fixed points often used in dynamics to infer under favorable
circumstances the existence of infinitely many periodic orbits.
There is no known single unifying explanation for the ubiquity of
Hamiltonian systems with infinitely many periodic orbits. Even in
the cases where the Conley conjecture holds, this is usually a
result of several disparate phenomena.

The survey can be read at three levels. First of all, we give a
broad picture, explain the main ideas, results, and conjectures in a
non-technical way, paying attention not only to what has been proved
but also to what is not known. This side of the survey requires very
little background in symplectic and contact topology and dynamics
from the reader. However, we also give the necessary technical
details and conditions when stating the most important results.
Although we recall the relevant definitions in due course, this
level of the survey is intended for a more expert reader. Finally,
in several instances, we attempt to explain the main ideas of the
proofs or even to sketch the arguments. In particular, in Sect.
3 we outline the proof of the Conley conjecture; here we
freely use Floer homology and some other, not entirely standard,
symplectic topological tools.

The survey is organized as follows. In Sect. 2, we
discuss the Conley conjecture (its history, background, and the
state of the art) and the generic existence results, and also set
the conventions and notation used throughout the paper. A detailed
outline of the proof is, as has been mentioned above, given in
Sect. 3. The rest of the paper is virtually independent
of this section. We discuss the Conley conjecture and other related
phenomena for Reeb flows in Sect. 4 and applications
of the contact Conley conjecture to twisted geodesic flows, which
govern the motion of a charge in a magnetic field, in Sect.
5. Finally, in Sect. 6, we turn to
the manifolds for which the Conley conjecture fails and, taking the
celebrated Frank’s theorem (see [Franks1992]; [Franks1996]) as a starting
point, show how certain local geometrical features of a system can
force the existence of infinitely many periodic orbits. Here we also
briefly touch upon the problem of the existence of infinitely many
simple periodic orbits for symplectomorphisms and for some other
types of “Hamiltonian”systems.

As has been pointed out in the introduction, for many closed
symplectic manifolds, every Hamiltonian diffeomorphism has
infinitely many simple periodic orbits and, in fact, simple periodic
orbits of arbitrarily large period whenever the fixed points are
isolated. This unconditional existence of infinitely many periodic
orbits is often referred to as the Conley conjecture. The conjecture
was, indeed, formulated by [Conley1984] for tori, and since then it has been a subject of active research
focusing on establishing the existence of infinitely many periodic
orbits for broader and broader classes of symplectic manifolds or
Hamiltonian diffeomorphisms.

The Conley conjecture was proved for the so-called weakly
non-degenerate Hamiltonian diffeomorphisms in [Salamon and Zehnder1992] (see also
[Conley and Zehnder1983b]) and for all Hamiltonian diffeomorphisms of
surfaces other than $S^{2}$ in [Franks and Handel2003] (see also [Le Calvez2006]). In its
original form for the tori, the conjecture was established in
[Hingston2009] (see also [Mazzucchelli2013]), and the case of an arbitrary
closed, symplectically aspherical manifold was settled in
[Ginzburg2010]. The proof was extended to rational, closed symplectic
manifolds $M$ with $c_{1}(TM)|_{\pi_{2}(M)}=0$ in [Ginzburg and Gürel2009b], and
the rationality requirement was then eliminated in [Hein2012]. In
fact, after [Salamon and Zehnder1992], the main difficulty in establishing the
Conley conjecture for more and more general manifolds with
aspherical first Chern class, overcome in this series of works, lied
in proving the conjecture for totally degenerate Hamiltonian
diffeomorphisms not covered by [Salamon and Zehnder1992]. (The internal logic in
[Franks and Handel2003] and [Le Calvez2006], relying on methods from low-dimensional dynamics, was
somewhat different.) Finally, in [Ginzburg and Gürel2012] and [Chance et al.2013], the
Conley conjecture was proved for negative monotone symplectic
manifolds. (The main new difficulty here was in the non-degenerate
case.)

Two other variants of the Hamiltonian Conley conjecture have also
been investigated. One of them is the existence of infinitely many
periodic orbits for Hamiltonian diffeomorphisms with displaceable
support; see, e.g., [Frauenfelder and Schlenk2007], [Gürel2008], [Hofer and Zehnder1994], [Schwarz2000] and [Viterbo1992]. Here the
form $\omega$ is usually required to be aspherical, but the manifold
$M$ is not necessarily closed. The second one is the Lagrangian
Conley conjecture or, more generally, the Conley conjecture for
Hamiltonians with controlled behavior at infinity on cotangent
bundles (see [Hein2011]; [Long2000]; [Lu2009]; [Lu2011]; [Mazzucchelli2011]) or even some
twisted cotangent bundles (see [Frauenfelder et al.2012]). In this survey, however,
we focus mainly on the case of closed manifolds.

The Conley conjecture looks deceptively similar to the well-known
conjecture that every closed simply connected Riemannian manifold
(e.g., $S^{n}$) carries infinitely many non-trivial closed geodesics.
However, this appears to be a very different problem than the Conley
conjecture, for the latter does not distinguish trivial and
non-trivial orbits. For instance, the proof of the Lagrangian Conley
conjecture for the pure kinetic energy Hamiltonian simply detects
the constant geodesics. We will further discuss the connection
between the two conjectures in Sects. 4
and 6.

What makes the Conley conjecture difficult and even counterintuitive
from a homological perspective is that there seems to be no obvious
homological reason for the existence of infinitely many simple
periodic orbits. As we have already mentioned, there is no
homological growth: the Floer homology of a Hamiltonian
diffeomorphism does not change under iterations and remains
isomorphic, up to a Novikov ring, to the homology of the manifold.
(In that sense, the difficulty is similar to that in proving
the existence of infinitely many non-trivial closed geodesics on,
say, $S^{n}$ where the rank of the homology of the free loop space
remains bounded as a function of the degree.)

Ultimately, one can expect the Conley conjecture to
hold for the majority of closed symplectic manifolds. There are,
however, notable exceptions. The simplest one is $S^{2}$: an
irrational rotation of $S^{2}$ about the $z$ axis has only two
periodic orbits, which are also the fixed points; these are the
poles. In fact, any manifold that admits a Hamiltonian torus action
with isolated fixed points also admits a Hamiltonian diffeomorphism
with finitely many periodic orbits. For instance, such a
diffeomorphism is generated by a generic element of the torus. In
particular, flag manifolds (hence the complex projective spaces and
the Grassmannians), and, more generally, most of the coadjoint
orbits of compact Lie groups as well as symplectic toric manifolds
all admit Hamiltonian diffeomorphisms with finitely many periodic
orbits. In dimension two, there are also such examples with
interesting dynamics. Namely, there exist area preserving
diffeomorphisms of $S^{2}$ with exactly three ergodic measures: two
fixed points and the area form; see [Anosov and Katok1970] and, e.g., [Fayad and Katok2004].
These are the so-called pseudo-rotations. By taking direct products
of pseudo-rotations, one obtains Hamiltonian diffeomorphisms of the
products of several copies of $S^{2}$ with finite number of ergodic
measures, and hence with finitely many periodic orbits. It would be
extremely interesting to construct a Hamiltonian analog of
pseudo-rotations for, say, ${\mathbb{CP}}^{2}$.

In all known examples of Hamiltonian diffeomorphisms with finitely
many periodic orbits, all periodic orbits are fixed points, i.e., no
new orbits are created by passing to the iterated diffeomorphisms,
cf. Sect. 6. Furthermore, all such Hamiltonian
diffeomorphisms are non-degenerate, and the number of fixed points
is exactly equal to the sum of Betti numbers. Note also that
Hamiltonian diffeomorphisms with finitely many periodic orbits are
extremely non-generic; see [Ginzburg and Gürel2009c] and
Sect. 2.2.

In any event, the class of manifolds admitting “counterexamples”
to the Conley conjecture appears to be very narrow, which leads one
to the question of finding further sufficient conditions for the
Conley conjecture to hold. There are several hypothetical
candidates. One of them, conjectured by the second author of this
paper, is that the minimal Chern number $N$ of $M$ is sufficiently
large, e.g., $N>\dim M$. (The condition $c_{1}(TM)|_{\pi_{2}(M)}=0$
corresponds to $N=\infty$.) More generally, it might be sufficient
to require the Gromov–Witten invariants of $M$ to vanish, as
suggested by Michael Chance and Dusa McDuff, or even the quantum
product to be undeformed. No results in these directions have been
proved to date. Note also that for all known “counterexamples” to
the Conley conjecture $\operatorname{H}_{*}(M;{\mathbb{Z}})$ is concentrated in even degrees.

Another feature of Hamiltonian diffeomorphisms with finitely many
periodic orbits is that, for many classes of manifolds, the actions
or the actions and the mean indices of their simple periodic orbits
must satisfy certain resonance relations of Floer homological
nature; see [Chance et al.2013], [Ginzburg and Gürel2009b], [Ginzburg and Kerman2010] and [Kerman2012]. (There are also
analogs of such resonance relations for Reeb flows, which we will
briefly touch upon in Sect. 4.) Although the very
existence of homological resonance relations in the Hamiltonian
setting is an interesting, new and unexpected phenomenon, and some
of the results considered here do make use of these relations, their
discussion is outside the scope of this paper.

In this section, we briefly introduce our basic conventions and
notation and then state the most up-to-date results on the Conley
conjecture and generic existence of infinitely many periodic orbits
for Hamiltonian diffeomorphisms.

Let us first recall the relevant terminology, some of which have
already been used in the previous section. A closed symplectic
manifold $(M^{2n},\omega)$ is said to be monotone (negative
monotone) if $[\omega]|_{\pi_{2}(M)}=\lambda c_{1}(TM)|_{\pi_{2}(M)}$ for
some non-negative (respectively, negative) constant $\lambda$ and
rational if $\left<[\omega],{\pi_{2}(M)}\right>=\lambda_{0}{\mathbb{Z}}$,
i.e., the integrals of $\omega$ over spheres in $M$ form a discrete
subgroup of ${\mathbb{R}}$. The positive generator $N$ of the discrete
subgroup $\left<c_{1}(TM),\pi_{2}(M)\right>\subset{\mathbb{R}}$ is called the
minimal Chern number of $M$. When this subgroup is zero, we
set $N=\infty$. A manifold $M$ is called symplectic CY
(Calabi–Yau) if $c_{1}(M)|_{\pi_{2}(M)}=0$ and symplectically
aspherical if $c_{1}(TM)|_{\pi_{2}(M)}=0=[\omega]|_{\pi_{2}(M)}$. A
symplectically aspherical manifold is monotone, and a monotone or
negative monotone manifold is rational.

All Hamiltonians $H$ considered in this paper are assumed to be
$k$-periodic in time (i.e., $H$ is a function $S^{1}_{k}\times M\to{\mathbb{R}}$,
where $S^{1}_{k}={\mathbb{R}}/k{\mathbb{Z}}$) and the period $k$ is always a positive
integer. When the period $k$ is not specified, it is equal to one,
and $S^{1}={\mathbb{R}}/{\mathbb{Z}}$. We set $H_{t}=H(t,\cdot)$ for $t\in S^{1}_{k}$. The
(time-dependent) Hamiltonian vector field $X_{H}$ of $H$ is defined by
$i_{X_{H}}\omega=-dH$. A Hamiltonian diffeomorphism is the
time-one map, denoted by $\varphi_{H}$ or just $\varphi$, of the
time-dependent Hamiltonian flow (i.e., Hamiltonian isotopy)
$\varphi_{H}^{t}$ generated by $X_{H}$. It is preferable throughout this
section to view $\varphi$ as an element, determined by
$\varphi_{H}^{t}$, of the universal covering of the group of Hamiltonian
diffeomorphisms. A one-periodic Hamiltonian $H$ can also be treated
as $k$-periodic. In this case, we will use the notation $H^{{\natural}k}$
and, abusing terminology, call $H^{{\natural}k}$ the $k$th iteration of
$H$.

In what follows, we identify the periodic orbits of $H$ (i.e., of
$\varphi_{H}^{t}$) with integer period $k$ and periodic orbits of
$\varphi$. A periodic orbit $x$ of $H$ is non-degenerate if
the linearized return map $d\varphi|_{x}\colon T_{x(0)}M\to T_{x(0)}M$ has no eigenvalues equal to one. Following [Salamon and Zehnder1992], we
call $x$ weakly non-degenerate if at least one of the
eigenvalues is different from one and totally degenerate
otherwise. Finally, a periodic orbit is said to be strongly
non-degenerate if no roots of unity are among the eigenvalues of
$d\varphi|_{x}$. This terminology carries over to Hamiltonians $H$
and Hamiltonian diffeomorphisms $\varphi$. For instance, $\varphi$
is non-degenerate if all its one-periodic orbits are non-degenerate
and strongly non-degenerate if all iterations $\varphi^{k}$ are
non-degenerate, etc.

The following theorem is the most general variant of the Conley
conjecture proved to date.

Theorem 2.1.

(Conley Conjecture) Assume that $M$ is a closed symplectic
manifold satisfying one of the following conditions:

(CY)

$c_{1}(TM)|_{\pi_{2}(M)}=0$
,

(NM)

$M$
is negative monotone.

Then every Hamiltonian diffeomorphism $\varphi$ of $M$ with finitely
many fixed points has simple periodic orbits of arbitrarily large
period.

As an immediate consequence, every Hamiltonian diffeomorphism of
$M$, whether or not the fixed-point set is finite, satisfying either
(CY) or (NM) has infinitely many simple periodic orbits. In fact,
when the fixed points of $\varphi$ are isolated, one can be even
more specific: if (CY) holds, every sufficiently large prime $p$
occurs as the period of a simple orbit and, moreover, one can show
that there exists a sequence of integers $l_{i}\to\infty$ such that
all $p^{l_{i}}$ are periods of simple orbits. Consequently, the number
of integers less than $k$ that occur as periods of simple periodic
orbits grows at least as fast as (in fact, faster than) $k/\ln k-C$
for some constant $C$. This lower growth bound is typical for the
Conley conjecture type results; see also [Ginzburg et al.2014] and Sect.
4 for the case of Reeb flows, and [Hingston1993] for
the growth of closed geodesics on $S^{2}$. (In dimension two, however,
stronger growth results have been established in some cases; see,
e.g., [Le Calvez2006]; Viterbo [Viterbo1992], Prop. 4.13 and also [Bramham and Hofer2012]; [Franks and Handel2003]; [Kerman2012].) When $M$ is negative monotone, it is only known that
there is a sequence of arbitrarily large primes occurring as simple
periods at least when, in addition, $\varphi$ is assumed to be
weakly non-degenerate; see Sect. 3.1.1 for the
definition.

The (CY) case of the theorem is proved in [Hein2012]; see also
[Ginzburg2010] and, respectively, [Ginzburg and Gürel2009b] for the proofs when
$M$ is symplectically aspherical, and when $M$ is rational and (CY)
holds. The negative monotone case is established in [Chance et al.2013] and [Ginzburg and Gürel2012]. For both classes of the ambient manifolds, the proof of
Theorem2.1 amounts to analyzing two cases. When $M$ is CY,
the “non-degenerate case” of the Conley conjecture is based on the
observation, going back to [Salamon and Zehnder1992], that unless $\varphi$ has a
fixed point of a particular type called a symplectically
degenerate maximum or an SDM, new simple periodic orbits of
high period must be created to generate the Floer homology in degree
$n=\dim M/2$. (For negative monotone manifolds, the argument is more
involved.) In the “degenerate case” one shows that the presence of
an SDM fixed point implies the existence of infinitely many periodic
orbits; see [Hingston2009] and also [Ginzburg2010]. We outline the proof of
Theorem2.1 for rational CY manifolds in Sect.
3.

Among closed symplectic manifolds $M$ with $c_{1}(TM)|_{\pi_{2}(M)}=0$
are tori and Calabi–Yau manifolds. In fact, the manifolds meeting
this requirement are more numerous than it might seem. As is proved
in [Fine and Panov2013], for every finitely presented group $G$ there exists a
closed symplectic 6-manifold $M$ with $\pi_{1}(M)=G$ and $c_{1}(TM)=0$.
A basic example of a negative monotone symplectic manifold is a
smooth hypersurface of degree $d>n+2$ in ${\mathbb{CP}}^{n+1}$. More
generally, a transverse intersection $M$ of $m$ hypersurfaces of
degrees $d_{1},\ldots,d_{m}$ in ${\mathbb{CP}}^{n+m}$ is negative monotone iff
$d_{1}+\cdots+d_{m}>n+m+1$; see Lawson and Michelsohn ([Lawson and Michelsohn1989], p. 88) and also, for $n=4$,
(McDuff and Salamon [McDuff and Salamon2004], pp. 429–430). A complete intersection $M$ is CY when
$d_{1}+\cdots+d_{m}=n+m+1$ and (strictly) monotone when $d_{1}+\cdots+d_{m}<n+m+1$. Thus “almost all” complete intersections are negative
monotone. Note also that the product of a symplectically aspherical
manifold and a negative monotone manifold is again negative
monotone.

As has been pointed out in Sect. 2.1, we expect
an analog of the theorem to hold when $N$ is large. [In the (CY)
case, $N=\infty$.] However, at this stage it is only known that the
number of simple periodic orbits is bounded from below by $\lceil N/n\rceil$ when $M^{2n}$ is rational and $2N>3n$; see Ginzburg and Gürel ([Ginzburg and Gürel2009b], Thm. 1.3).

Let us now turn to the question of the generic existence of
infinitely many simple periodic orbits. Conjecturally, for any
closed symplectic manifold $M$, a $C^{\infty}$-generic Hamiltonian
diffeomorphism has infinitely many simple periodic orbits . This,
however, is unknown (somewhat surprisingly) and appears to be a
non-trivial problem. In all results to date, some assumptions on $M$
are required for the proof.

Theorem 2.2.

(Generic existence) Assume that $M^{2n}$ is a closed
symplectic manifold with minimal Chern number $N$, meeting one of
the following requirements:

(i)

$\operatorname{H}_{\textrm{odd}}(M;R)\neq 0$
for some ring
$R$
,

(ii)

$N\geq n+1$
,

(iii)

$M$
is
${\mathbb{CP}}^{n}$
or a complex Grassmannian or a product
of one of these manifolds with a closed symplectically aspherical
manifold.

Then strongly non-degenerate Hamiltonian diffeomorphisms with
infinitely many simple periodic orbits form a $C^{\infty}$-residual
set in the space of all $C^{\infty}$-smooth Hamiltonian
diffeomorphisms.

This theorem is proved in [Ginzburg and Gürel2009c]. In (iii), instead of
explicitly specifying $M$, we could have required that $M$ is
monotone and that there exists $u\in\operatorname{H}_{*<2n}(M)$ with $2n-\deg u<2N$ and $w\in\operatorname{H}_{*<2n}(M)$ and $\alpha$ in the Novikov ring of
$M$ such that $[M]=(\alpha u)*w$ in the quantum homology. We refer
the reader to [Ginzburg and Gürel2009c] for other examples when this
condition is satisfied and a more detailed discussion.

The proof of the theorem when (i) holds is particularly simple.
Namely, in this case, a non-degenerate Hamiltonian diffeomorphism
$\varphi$ with finitely many periodic orbits must have a
non-hyperbolic periodic orbit. Indeed, it follows from Floer theory
that $\varphi$ has a non-hyperbolic fixed point or a hyperbolic
fixed point with negative eigenvalues. When $\varphi$ has finitely
many periodic orbits, we can eliminate the latter case by passing to
an iteration of $\varphi$. To finish the proof it suffices to apply
the Birkhoff–Lewis–Moser theorem, [Moser1977]. (This argument is
reminiscent of the reasoning in, e.g., [Markus and Meyer1980] where the generic
existence of solenoids for Hamiltonian flows is established.) The
proofs of the remaining cases rely on the fact, already mentioned in
Sect. 2.1, that under our assumptions on $M$ the
indices and/or actions of the periodic orbits of $\varphi$ must
satisfy certain resonance relations when $\varphi$ has only finitely
many periodic orbits; see [Ginzburg and Gürel2009b] and [Ginzburg and Kerman2010]. These resonance
relations can be easily broken by a $C^{\infty}$-small perturbation of
$\varphi$, and the theorem follows.

It is interesting to look at these results in the context of the
closing lemma, which implies that the existence of a dense set of
periodic orbits is $C^{1}$-generic for Hamiltonian diffeomorphisms;
see [Pugh and Robinson1983]. Thus, once the $C^{\infty}$-topology is replaced by the
$C^{1}$-topology a much stronger result than the generic existence of
infinitely many periodic orbits holds—the generic dense
existence. However, this is no longer true for the $C^{r}$-topology
with $r>\dim M$ as the results of M. Herman show (see [Herman1991a]; [Herman1991b]), and the above conjecture on the $C^{\infty}$-generic
existence of infinitely many periodic orbits can be viewed as a
hypothetical variant of a $C^{\infty}$-closing lemma. (Note also that
in the closing lemma one can require the perturbed diffeomorphism to
be $C^{\infty}$-smooth, but only $C^{1}$-close to the original one, as
long as only finitely many periodic orbits are created. It is not
clear to us whether one can produce infinitely many periodic orbits
by a $C^{\infty}$-smooth $C^{1}$-small perturbation.)

An interesting consequence of Theorem2.2 pointed out
in [Polterovich and Shelukhin2014] is that non-autonomous Hamiltonian diffeomorphisms
(i.e., Hamiltonian diffeomorphisms that cannot be generated by
autonomous Hamiltonians) on a manifold meeting the conditions of the
theorem form a $C^{\infty}$-residual subset in the space of all
$C^{\infty}$-smooth Hamiltonians. Indeed, when $k>1$, simple
$k$-periodic orbits of an autonomous Hamiltonian diffeomorphism are
never isolated, and hence, in particular, never non-degenerate.

Remark 2.3.

The proofs of Theorem2.2 utilize
Hamiltonian Floer theory. Hence, either $M$ is required in addition
to be weakly monotone [i.e., $M$ is monotone or $N>n-2$; see
([Hofer and Salamon1995]; [McDuff and Salamon2004]; [Ono1995]; [Salamon1999]) for more details] or the proofs
ultimately, although not explicitly, must rely on the machinery of
multi-valued perturbations and virtual cycles [see ([Fukaya and Ono1999]; [Fukaya et al.2009]; [Liu and Tian1998]) or, for the polyfold approach, ([Hofer et al.2010]; [Hofer et al.2011]) and
references therein]. In the latter case, the ground field in the
Floer homology must have zero characteristic.

To every continuous path $\Phi\colon[0,\,1]\to\operatorname{Sp}(2n)$ starting at
$\Phi(0)=I$ one can associate the mean index $\Delta(\Phi)\in{\mathbb{R}}$, a homotopy invariant of the path with fixed end-points; see
[Long2002] and [Salamon and Zehnder1992]. To give a formal definition, recall first that a map
$\Delta$ from a Lie group to ${\mathbb{R}}$ is said to be a quasimorphism if
it fails to be a homomorphism only up to a constant, i.e.,
$|\Delta(\Phi\Psi)-\Delta(\Phi)-\Delta(\Psi)|<{const}$, where the
constant is independent of $\Phi$ and $\Psi$. One can prove that
there is a unique quasimorphism $\Delta\colon\widetilde{\operatorname{Sp}}(2n)\to{\mathbb{R}}$ which is continuous and homogeneous [i.e.,
$\Delta(\Phi^{k})=k\Delta(\Phi)$] and satisfies the normalization
condition: $\Delta(\Phi_{0})=2$ for $\Phi_{0}(t)=e^{2\pi it}\oplus I_{2n-2}$ with $t\in[0,\,1]$, in the self-explanatory notation; see
[Barge and Ghys1992]. This quasimorphism is the mean index. (The continuity
requirement holds automatically and is not necessary for the
characterization of $\Delta$, although this is not immediately
obvious. Furthermore, $\Delta$ is also automatically conjugation
invariant, as a consequence of the homogeneity.)

The mean index $\Delta(\Phi)$ measures the total rotation angle of
certain unit eigenvalues of $\Phi(t)$ and can be explicitly defined
as follows. For $A\in\operatorname{Sp}(2)$, set $\rho(A)=e^{i\lambda}\in S^{1}$ when
$A$ is conjugate to the rotation by $\lambda$ counterclockwise,
$\rho(A)=e^{-i\lambda}\in S^{1}$ when $A$ is conjugate to the rotation
by $\lambda$ clockwise, and $\rho(A)=\pm 1$ when $A$ is hyperbolic
with the sign determined by the sign of the eigenvalues of $A$. Then
$\rho\colon\operatorname{Sp}(2)\to S^{1}$ is a continuous (but not $C^{1}$) function,
which is conjugation invariant and equal to $\det$ on $\operatorname{U}(1)$. A
matrix $A\in\operatorname{Sp}(2n)$ with distinct eigenvalues, can be written as
the direct sum of matrices $A_{j}\in\operatorname{Sp}(2)$ and a matrix with complex
eigenvalues not lying on the unit circle. We set $\rho(A)$ to be the
product of $\rho(A_{j})\in S^{1}$. Again, $\rho$ extends to a continuous
function $\rho\colon\operatorname{Sp}(2n)\to S^{1}$, which is conjugation invariant
(and hence $\rho(AB)=\rho(BA)$) and restricts to $\det$ on $\operatorname{U}(n)$;
see, e.g., [Salamon and Zehnder1992]. Finally, given a path $\Phi\colon[0,\,1]\to\operatorname{Sp}(2n)$, there is a continuous function $\lambda(t)$ such that
$\rho(\Phi(t))=e^{i\lambda(t)}$, measuring the total rotation of the
“preferred” eigenvalues on the unit circle, and we set
$\Delta(\Phi)=(\lambda(1)-\lambda(0))/2$.

Assume now that the path $\Phi$ is non-degenerate, i.e., by
definition, all eigenvalues of the end-point $\Phi(1)$ are different
from one. We denote the set of such matrices in $\operatorname{Sp}(2n)$ by
$\operatorname{Sp}^{*}(2n)$. It is not hard to see that $\Phi(1)$ can be connected
to a hyperbolic symplectic transformation by a path $\Psi$ lying
entirely in $\operatorname{Sp}^{*}(2n)$. Concatenating this path with $\Phi$, we
obtain a new path $\Phi^{\prime}$. By definition, the Conley–Zehnder
index ${{\mu}_{{\rm CZ}}}(\Phi)\in{\mathbb{Z}}$ of $\Phi$ is $\Delta(\Phi^{\prime})$. One can show
that ${{\mu}_{{\rm CZ}}}(\Phi)$ is well-defined, i.e., independent of $\Psi$.
Furthermore, following [Salamon and Zehnder1992], let us call $\Phi$ weakly
non-degenerate if at least one eigenvalue of $\Phi(1)$ is different
from one and totally degenerate otherwise. The path is
strongly non-degenerate if all its “iterations” $\Phi^{k}$
are non-degenerate, i.e., none of the eigenvalues of $\Phi(1)$ is a
root of unity.

The indices $\Delta$ and ${{\mu}_{{\rm CZ}}}$ have the following properties:

(CZ1)

$|\Delta(\Phi)-{{\mu}_{{\rm CZ}}}(\tilde{\Phi})|\leq n$ for every
sufficiently small non-degenerate perturbation $\tilde{\Phi}$ of $\Phi$;
moreover, the inequality is strict when $\Phi$ is weakly
non-degenerate.

(CZ2)

${{\mu}_{{\rm CZ}}}(\Phi^{k})/k\to\Delta(\Phi)$ as $k\to\infty$, when
$\Phi$ is strongly non-degenerate; hence the name “mean index” for
$\Delta$.

Note that with our conventions the Conley–Zehnder index of a path
parametrized by $[0,\,1]$ and generated by a small negative definite
quadratic Hamiltonian on ${\mathbb{R}}^{2n}$ is $n$.

Let now $M^{2n}$ be a symplectic manifold and $x\colon S^{1}\to M$ be
a contractible loop. A capping of $x$ is a map $u\colon D^{2}\to M$ such that $u|_{S^{1}}=x$. Two cappings $u$ and $v$ of $x$
are considered to be equivalent if the integrals of $c_{1}(TM)$ and
$\omega$ over the sphere $u\#v$ obtained by attaching $u$ to $v$
are equal to zero. A capped closed curve $\bar{x}=(x,u)$ is, by
definition, a closed curve $x$ equipped with an equivalence class of
a capping. In what follows, a capping is always indicated by the
bar.

For a capped one-periodic (or $k$-periodic) orbit $\bar{x}$ of a
Hamiltonian $H\colon S^{1}\times M\to{\mathbb{R}}$, we can view the linearized
flow $d\varphi_{H}^{t}|_{x}$ along $x$ as a path in $\operatorname{Sp}(2n)$ by fixing a
trivialization of $u^{*}TM$ and restricting it to $x$. With this
convention in mind, the above definitions and constructions apply to
$\bar{x}$ and, in particular, we have the mean index $\Delta(\bar{x})$ and,
when $x$ is non-degenerate, the Conley–Zehnder index ${{\mu}_{{\rm CZ}}}(\bar{x})$
defined. These indices are independent of the trivialization of
$u^{*}TM$, but may depend on the capping. Furthermore, (CZ1) and
(CZ2) hold. The difference of the indices of $(x,u)$ and
$(x,v)$ is equal to $2\left<c_{1}(TM),u\#v\right>$. Hence, when $M$
is a symplectic CY manifold, the indices are independent of the
capping and thus assigned to $x$. The terminology we introduced for
paths in $\operatorname{Sp}(2n)$ translates word-for-word to periodic orbits and
Hamiltonian diffeomorphisms, cf. Sect. 2.2.1.

The space of capped closed curves is a covering space of the space
of contractible loops, and the critical points of ${\mathcal{A}}_{H}$ on this
covering space are exactly capped one-periodic orbits of $X_{H}$. The
action spectrum ${\mathcal{S}}(H)$ of $H$ is the set of critical values
of ${\mathcal{A}}_{H}$. This is a zero measure set; see, e.g., [Hofer and Zehnder1994]. When
$M$ is rational, ${\mathcal{S}}(H)$ is a closed and hence nowhere dense set.
[Otherwise, ${\mathcal{S}}(H)$ is dense.] These definitions extend to
$k$-periodic orbits and Hamiltonians in the obvious way. Clearly,
the action functional is homogeneous with respect to iteration:

Here $\bar{x}^{k}$ stands for the $k$th iteration of the capped orbit
$\bar{x}$.

For a Hamiltonian $H\colon S^{1}\times M\to{\mathbb{R}}$ and
$\varphi=\varphi_{H}$, we denote by $\operatorname{HF}_{*}(\varphi)$ or, when the
action filtration is essential, by $\operatorname{HF}_{*}^{(a,\,b)}(H)$ the
Floer homology of $H$, where $a$ and $b$ are not in ${\mathcal{S}}(H)$.
We refer the reader to, e.g., [McDuff and Salamon2004] and [Salamon1999] for a detailed
construction of the Floer homology and to [Ginzburg and Gürel2009b] for a
treatment particularly tailored for our purposes. Here we only
mention that, when $H$ is non-degenerate, $\operatorname{HF}_{*}^{(a,\,b)}(H)$ is
the homology of a complex generated by the capped one-periodic
orbits of $H$ with action in the interval $(a,\,b)$ and graded by
the Conley–Zehnder index. Furthermore, $\operatorname{HF}_{*}(\varphi)\cong\operatorname{H}_{*+n}(M)\otimes\Lambda$, where $\Lambda$ is a suitably defined
Novikov ring. As a consequence, $\operatorname{HF}_{n}(\varphi)\neq 0$ when $M$ is
symplectic CY. (For our purposes it is sufficient to take ${\mathbb{Z}}_{2}$ as
the ground field.)

When $x$ is an isolated one-periodic orbit of $H$, one can associate
to it the so-called local Floer homology $\operatorname{HF}_{*}(x)$ of $x$.
This is the homology of a complex generated by the orbits $x_{i}$
which $x$ splits into under a $C^{\infty}$-small non-degenerate
perturbation. The differential $\partial$ is defined similarly to the
standard Floer differential, and to show that $\partial^{2}=0$ it suffices
to prove that the Floer trajectories connecting the orbits $x_{i}$
cannot approach the boundary of an isolating neighborhood of $x$.
This is an immediate consequence of Floer ([Floer1989], Thm. 3); see also
[McLean2012] for a different proof. The resulting homology is well
defined, i.e., independent of the perturbation. The local Floer
homology $\operatorname{HF}_{*}(x)$ carries only a relative grading. To have a
genuine ${\mathbb{Z}}$-grading it is enough to fix a trivialization of
$TM|_{x}$. In what follows, such a trivialization will usually come
from a capping of $x$, and we will then write $\operatorname{HF}_{*}(\bar{x})$. Clearly,
the grading is independent of the capping when
$c_{1}(TM)|_{\pi_{2}(M)}=0$. Hence, in the symplectic (CY) case, the
local Floer homology is associated to the orbit $x$ itself. With
relative grading, the local Floer homology is defined for the germ
of a time-dependent Hamiltonian flow or, when $x$ is treated as a
fixed point, of a Hamiltonian diffeomorphism. The local Floer
homology is invariant under deformations of $H$ as long as $x$ stays
uniformly isolated.

Example 3.1.

When $x$ is non-degenerate, $\operatorname{HF}_{*}(\bar{x})\cong{\mathbb{Z}}_{2}$ is concentrated in
degree ${{\mu}_{{\rm CZ}}}(\bar{x})$. When $x$ is an isolated critical point of an
autonomous $C^{2}$-small Hamiltonian $F$ (with trivial capping), the
local Floer homology is isomorphic to the local Morse homology
$\operatorname{HM}_{*+n}(F,x)$ of $F$ at $x$ (see [Ginzburg2010]), also known as
critical modules, which is in turn isomorphic to
$\operatorname{H}_{*}(\{F<c\}\cup\{x\},\{F<c\})$, where $F(x)=c$. The isomorphism
$\operatorname{HF}_{*}(x)\cong\operatorname{HM}_{*+n}(F,x)$ is a local analog of the isomorphism
between the Floer and Morse homology groups of a $C^{2}$-small
Hamiltonian; see [Salamon and Zehnder1992] and references therein.

Let us now state three properties of local Floer homology, which are
essential for what follows.

First of all, $\operatorname{HF}_{*}(\bar{x})$ is supported in the interval
$[\Delta(\bar{x})-n,\Delta(\bar{x})+n]$:

i.e., the homology vanishes in the degrees outside this interval.
Moreover, when $x$ is weakly non-degenerate, the support lies in the
open interval. These facts readily follow from (CZ1) and the
continuity of the mean index.

Secondly, the local Floer homology groups are building blocks for
the ordinary Floer homology. Namely, assume that for $c\in{\mathcal{S}}(H)$
there are only finitely many one-periodic orbits $\bar{x}_{i}$ with
${\mathcal{A}}_{H}(\bar{x}_{i})=c$. Then all these orbits are isolated and

when $M$ is rational and $\epsilon>0$ is sufficiently small.
Furthermore, it is easy to see that, even without the rationality
condition, $\operatorname{HF}_{l}(\varphi)=0$ when all one-periodic orbits of $H$
are isolated and have local Floer homology vanishing in degree $l$.

Finally, the local Floer homology enjoys a certain periodicity
property as a function of the iteration order. To be more specific,
let us call a positive integer $k$ an admissible iteration of
$x$ if the multiplicity of the generalized eigenvalue one for the
iterated linearized Poincaré return map $d\varphi^{k}|_{x}$ is equal
to its multiplicity for $d\varphi|_{x}$. In other words, $k$ is
admissible if and only if it is not divisible by the degree of any
root of unity different from one among the eigenvalues of
$d\varphi|_{x}$. For instance, when $x$ is totally degenerate (the
only eigenvalue is one) or strongly non-degenerate (no roots of
unity among the eigenvalues), all $k\in{\mathbb{N}}$ are admissible. For any
$x$, all sufficiently large primes are admissible. We have

Theorem 3.2.

([Ginzburg and Gürel2010]) Let $\bar{x}$ be a capped isolated one-periodic
orbit of a Hamiltonian $H\colon S^{1}\times M\to{\mathbb{R}}$. Then $x^{k}$ is
also an isolated one-periodic orbit of $H^{{\natural}k}$ for all
admissible $k$, and the local Floer homology groups of $\bar{x}$ and
$\bar{x}^{k}$ coincide up to a shift of degree:

Furthermore, $\lim_{k\to\infty}s_{k}/k=\Delta(\bar{x})$ and
$s_{k}=k\Delta(\bar{x})$ for all $k$ when $x$ is totally degenerate.
Moreover, when $\operatorname{HF}_{n+\Delta(\bar{x})}(\bar{x})\neq 0$, the orbit $x$ is
totally degenerate.

The first part of this theorem is an analog of the result from
[Gromoll and Meyer1969] for Hamiltonian diffeomorphisms. One can replace a capping
of $x$ by a trivialization of $TM|_{x}$ with the grading and indices
now associated with that trivialization. The theorem is not obvious,
although not particularly difficult. First, by using a variant of
the Künneth formula and some simple tricks, one can reduce the
problem to the case where $x$ is a totally degenerate constant orbit
with trivial capping. [Hence, in particular, $\Delta(\bar{x})=0$]. Then
we have the isomorphisms $\operatorname{HF}_{*}(\bar{x})=\operatorname{HM}_{*+n}(F,x)$, where $F\colon M\to{\mathbb{R}}$, near $x$, is the generating function of $\varphi$, and
$\operatorname{HF}_{*}(\bar{x}^{k})=\operatorname{HM}_{*+n}(kF,x)=\operatorname%
{HM}_{*+n}(F,x)$. Thus, in the totally
degenerate case, $s_{k}=0$, and the theorem follows; see [Ginzburg and Gürel2010]
for a complete proof. [The fact that $x^{k}$ is automatically isolated
when $k$ is admissible, reproved in [Ginzburg and Gürel2010], has been known
for some time; see [Chow et al.1981].]

As a consequence of Theorem3.2 or of [Chow et al.1981],
the iterated orbit $x^{k}$ is automatically isolated for all $k$ if it
is isolated for some finite collection of iterations $k$ (depending
on the degrees of the roots of unity among the eigenvalues).
Furthermore, it is easy to see that then the map $k\mapsto\operatorname{HF}_{*}(\bar{x}^{k})$ is periodic up to a shift of grading, and hence the
function $k\mapsto\dim\operatorname{HF}_{*}(\bar{x}^{k})$ is bounded.

An isolated orbit $x$ is said to be homologically non-trivial
if $\operatorname{HF}_{*}(x)\neq 0$. (The choice of trivialization along the orbit
is clearly immaterial here.) These are the orbits detected by the
filtered Floer homology. For instance, a non-degenerate orbit is
homologically non-trivial. By Theorem3.2, an
admissible iteration of a homologically non-trivial orbit is again
homologically non-trivial. It is not known if, in general, an
iteration of a homologically non-trivial orbit can become
homologically trivial while remaining isolated.

We refer the reader to [Ginzburg2010] and [Ginzburg and Gürel2009b]; [Ginzburg and Gürel2010] for a further
discussion of local Floer homology.

As we noted in Sect. 2.2, the proof of the
general case of the Conley conjecture for symplectic CY manifolds
hinges on the fact that the presence of an orbit of a particular
type, a symplectically degenerate maximum or an SDM,
automatically implies the existence of infinitely many simple
periodic orbits. To be more precise, an isolated periodic orbit $x$
is said to be a symplectically degenerate maximum if $\operatorname{HF}_{n}(x)\neq 0$ and $\Delta(x)=0$ for some trivialization. This definition makes
sense even for the germs of Hamiltonian flows or Hamiltonian
diffeomorphisms. An SDM orbit is necessarily totally degenerate by
the “moreover” part of (3.1)‣ or Theorem3.2. Sometimes it is also convenient to say that an
orbit is an SDM with respect to a particular capping. For instance,
a capped orbit $\bar{x}$ is an SDM if it is an SDM for the trivialization
associated with the capping, i.e., $\operatorname{HF}_{n}(\bar{x})\neq 0$ and
$\Delta(\bar{x})=0$.

Example 3.3.

Let $H\colon{\mathbb{R}}^{2n}\to{\mathbb{R}}$ be an autonomous Hamiltonian with an
isolated critical point at $x=0$. Assume furthermore that $x$ is a
local maximum and that all eigenvalues (in the sense of, e.g.,
[Arnold1989], App. 6) of the Hessian $d^{2}H(x)$ are equal to zero. Then
$x$ (with constant trivialization or, equivalently, trivial capping)
is an SDM of $H$. For instance, the origin in ${\mathbb{R}}^{2}$ is an SDM for
$H(p,q)=p^{4}+q^{4}$ or $H(p,q)=p^{2}+q^{4}$, but not for $H(p,q)=ap^{2}+bq^{2}$
for any $a\neq 0$ and $b\neq 0$.

Remark 3.4.

There are several other ways to define an SDM. The following
conditions are equivalent (see [Ginzburg and Gürel2010], Prop. 5.1):

•

the orbit
$\bar{x}$
is a symplectically degenerate maximum of
$H$
;

•

$\operatorname{HF}_{n}(\bar{x}^{k_{i}})\neq 0$
for some sequence of admissible
iterations
$k_{i}\to\infty$
;

•

the orbit
$x$
is totally degenerate,
$\operatorname{HF}_{n}(\bar{x})\neq 0$
and
$\operatorname{HF}_{n}(\bar{x}^{k})\neq 0$
for at least one admissible iteration
$k\geq n+1$
.

The following proposition settling, in particular, the
non-degenerate case of the Conley conjecture for symplectic CY
manifolds is a refinement of the main result from [Salamon and Zehnder1992]. It is
proved in [Ginzburg2010] and [Ginzburg and Gürel2009b], although the argument given below
is somewhat different from the original proof.

Proposition 3.5.

Assume that $c_{1}(TM)|_{\pi_{2}(M)}=0$ and that $\varphi=\varphi_{H}$ has
finitely many fixed points and none of these points is an SDM. (This
is the case when, e.g., $\varphi$ is weakly non-degenerate.) Then
$\varphi$ has a simple periodic orbit of period $k$ for every
sufficiently large prime $k$.

The key to the proof is the fact that $\operatorname{HF}_{n}(\varphi^{k})\neq 0$ for
all $k$ and that, even when $\omega|_{\pi_{2}(M)}\neq 0$, the
condition $c_{1}(TM)|_{\pi_{2}(M)}=0$ guarantees that all recappings of
every orbit have the same mean index and the same (graded) local
Floer homology.

Proof.

First, note that when $k$ is prime, every $k$-periodic orbit is
either simple or the $k$-th iteration of a fixed point. For every
isolated fixed point $x$, we have three mutually exclusive
possibilities:

•

$\Delta(x)\neq 0$,

•

$\Delta(x)=0$ and $\operatorname{HF}_{n}(x)=0$,

•

$\Delta(x)=0$ but $\operatorname{HF}_{n}(x)\neq 0$.

Here we are using the fact that $M$ is CY, and hence the indices are
independent of the capping. The last case, where $x$ is an SDM, is
ruled out by the assumptions of the proposition.

In the first case, $\operatorname{HF}_{n}(x^{k})=0$ when $k|\Delta(x)|>2n$, and hence
$x$ cannot contribute to $\operatorname{HF}_{n}(\varphi^{k})$ when $k$ is large. In
the second case, $\operatorname{HF}_{n}(x^{k})=\operatorname{HF}_{n}(x)=0$ for all admissible
iterations by Theorem3.2. In particular, $x$ again
cannot contribute to $\operatorname{HF}_{n}(\varphi^{k})$ for all large primes $k$. It
follows that, under the assumptions of the proposition,
$\operatorname{HF}_{n}(\varphi^{k})=0$ for all large primes $k$ unless $\varphi$ has a
simple periodic orbit of period $k$.
$\square$

Remark 3.6.

Although this argument relies on Theorem3.2 which
is not entirely trivial, a slightly different logical organization
of the proof would enable one to utilize a much simpler of version
of the theorem; see [Ginzburg2010] and [Ginzburg and Gürel2009b].

With Proposition3.5 established, it remains to deal
with the degenerate case of the Conley conjecture, i.e., the case
where $\varphi$ has an SDM. We do this in the next section; see
Theorem3.8.

Remark 3.7.

When $M$ is negative monotone and $\varphi$ has an SDM fixed point,
the degenerate case of Theorem2.1 follows from Theorem3.8, just as for CY manifolds. However, the
non-degenerate case requires a totally new proof. The argument
relies on the sub-additivity property of spectral invariants; see
[Chance et al.2013] and [Ginzburg and Gürel2012] for more details.

In this section, we show that a Hamiltonian diffeomorphism with an
SDM fixed point has infinitely many simple periodic orbits. We
assume that $M$ is rational as in [Ginzburg and Gürel2009b]. The case of
irrational CY manifolds is treated in [Hein2012].

Theorem 3.8.

([Ginzburg and Gürel2009b]) Let $\varphi=\varphi_{H}$ be a Hamiltonian
diffeomorphism of a closed rational symplectic manifold $M$,
generated by a one-periodic Hamiltonian $H$. Assume that some
iteration $\varphi^{k_{0}}$ has finitely many $k_{0}$-periodic orbits
and one of them, $\bar{x}$, is an SDM.

(i)

Then
$\varphi$
has infinitely many simple periodic
orbits.

(ii)

Moreover,
$\varphi$
has simple periodic orbits of
arbitrarily large prime period if, in addition,
$k_{0}=1$
and
$\omega|_{\pi_{2}(M)}=0$
or
$c_{1}(M)|_{\pi_{2}(M)}=0$
.

This theorem is in turn a consequence of the following result.

Theorem 3.9.

([Ginzburg and Gürel2009b]) Assume that $(M^{2n},\omega)$ is closed and
rational, and let $\bar{x}$ be an SDM of $H$. Set $c={\mathcal{A}}_{H}(\bar{x})$. Then
for every sufficiently small $\epsilon>0$ there exists $k_{\epsilon}$ such
that

$\operatorname{HF}^{(kc+\delta_{k},\,kc+\epsilon)}_{n+1}\big(H^{{\natural}k}%
\big)\neq 0\text{ for all }k>k_{\epsilon}\text{ and some }\delta_{k}\text{ %
with }0<\delta_{k}<\epsilon.$

(3.2)

For instance, to prove case (ii) of Theorem3.8
when $M$ is CY it suffices to observe that no $k$th iteration of a
fixed point can contribute to the Floer homology in degree $n+1$ for
any action interval when $k$ is a sufficiently large prime and
$\varphi$ has finitely many fixed points. When
$\omega|_{\pi_{2}(M)}=0$, the argument is similar, but now the action
filtration is used in place of the degree. The proof of case (i) is
more involved; see Ginzburg and Gürel ([Ginzburg and Gürel2009b], Sect. 3) where some more general
results are also established.

It is worth pointing out that although Theorem3.9
guarantees the existence of capped simple $k$-periodic orbits $\bar{y}$
with action close to $kc={\mathcal{A}}_{H^{{\natural}k}}(\bar{x}^{k})$, we do not claim
that the orbits $y$ are close to $x^{k}$ or even intersect a
neighborhood of $x$. In general, essentially nothing is known about
the location of these orbits and hypothetically a small neighborhood
of $x$ may contain no periodic orbits at all. However, as is proved
in [Yan2014], the orbit $x$ is in a certain sense an accumulation
point for periodic orbits when, e.g., $M={\mathbb{T}}^{2}$.

Outline of the proof of Theorem3.9 Composing if necessary $\varphi^{t}_{H}$ with a loop of Hamiltonian
diffeomorphisms, we can easily reduce the problem to the case where
$\bar{x}$ is a constant one-periodic orbit with trivial capping; see
Ginzburg and Gürel ([Ginzburg and Gürel2009b], Prop. 2.9 and 2.10). Henceforth, we write $x$ rather
than $\bar{x}$ and assume that $dH_{t}(x)=0$ for all $t$.

The key to the proof is the following geometrical characterization
of SDMs:

Lemma 3.10.

([Ginzburg2010]; [Hingston2009]) Let $x$ be an isolated constant one-periodic
orbit for a germ of a time-dependent Hamiltonian flow $\varphi_{H}^{t}$.
Assume that $x$ (with constant trivialization) is an SDM. Then there
exists a germ of a time-dependent Hamiltonian flow $\varphi_{K}^{t}$
near $x$ such that the two flows generate the same time-one map,
i.e., $\varphi_{K}=\varphi_{H}$, and $K_{t}$ has a strict local maximum at
$x$ for every $t$. Furthermore, one can ensure that the Hessian
$d^{2}K_{t}(x)$ is arbitrarily small. In other words, for every $\eta>0$
one can find such a Hamiltonian $K_{t}$ with $\|d^{2}K_{t}(x)\|<\eta$.

Remark 3.11.

Strictly speaking, contrary to what is stated in Ginzburg and Gürel ([Ginzburg and Gürel2009b], Rmk. 5.9, [Ginzburg and Gürel2010], Prop. 5.2), this lemma is not quite
a characterization of SDMs in the sense that it is not clear if
every $x$ for which such Hamiltonians $K_{t}$ exist is necessarily an
SDM. However, in fact, $K_{t}$ can be taken to meet an additional
requirement ensuring, in essence, that the $t$-dependence of $K_{t}$
is minor. With this condition, introduced in Hingston ([Hingston2009], Lemma 4) as
that $K$ is relatively autonomous (see also [Ginzburg2010], Sect. 5 and
6), the lemma gives a necessary and sufficient condition for
an SDM.

Outline of the proof of Lemma3.10.

The proof of Lemma3.10 is rather technical, but the idea
of the proof is quite simple. Set $\varphi=\varphi_{H}$. First, note
that all eigenvalues of $d\varphi|_{x}\colon T_{x}M\to T_{x}M$ are equal
to one since $x$ is totally degenerate. Thus by applying a
symplectic linear change of coordinates we can bring $d\varphi|_{x}$
arbitrarily close to the identity. Then $\varphi$ is also
$C^{1}$-close to ${id}$ near $x$. Let us identify $(M\times M,\omega\oplus(-\omega))$ near $(x,x)$ with a neighborhood of the
zero section in $T^{*}M$ near $x$, and hence the graph of $\varphi$
with the graph of $dF$ for a germ of a smooth function $F$ near $x$.
The function $F$ is a generating function of $\varphi$. Clearly, $x$
is an isolated critical point of $F$ and $d^{2}F(x)=O(\|d\varphi|_{x}-I\|)$.

Furthermore, similarly to Example3.1, we have an
isomorphism

$\operatorname{HF}_{*}(x)=\operatorname{HM}_{*+n}(F,x),$

and thus $\operatorname{HM}_{2n}(F,x)\neq 0$. It is routine to show that an
isolated critical point $x$ of a function $F$ is a local maximum if
and only if $\operatorname{HM}_{2n}(F,x)\neq 0$. The generating function $F$ is
not quite a Hamiltonian generating $\varphi$, but it is not hard to
turn $F$ into such a Hamiltonian $K_{t}$ and check that $K_{t}$ inherits
the properties of $F$.
$\square$

Returning to the proof of Theorem3.9, we apply the lemma
to the SDM orbit $x$ and observe that the local loop
$\varphi_{H}^{t}\circ(\varphi_{K}^{t})^{-1}$ has zero Maslov index and hence
is contractible. It is not hard to show that every local
contractible loop extends to a global contractible loop; see
Ginzburg ([Ginzburg2010], Lemma 2.8). In other words, we can extend the
Hamiltonian $K_{t}$ from Lemma3.10 to a global Hamiltonian
such that $\varphi_{K}=\varphi$, not only near $x$ but on the entire
manifold $M$.

With this in mind, let us reset the notation.
Replacing $H$ by $K$ but retaining the original notation, we can say
that for every $\eta>0$ there exists a Hamiltonian $H$ such that

•

$\varphi_{H}=\varphi$;

•

$x$ is a constant periodic orbit of $H$, and $H_{t}$ has an
isolated local maximum at $x$ for all $t$;

•

$\|d^{2}H_{t}(x)\|<\eta$ for all $t$.

Furthermore, we can always assume that all such Hamiltonians $H$ are
related to each other and to the original Hamiltonian via global
loops with zero action and zero Maslov index. Thus, in particular,
$c={\mathcal{A}}_{H}(x)=H(x)$ is independent of the choice of $H$ above, and all
Hamiltonians have the same filtered Floer homology. Therefore, it is
sufficient to prove the theorem for any of these Hamiltonians $H$
with arbitrarily small Hessian $d^{2}H(x)$.

To avoid technical difficulties and illuminate the idea of the
proof, let us asume that $d^{2}H(x)=0$ and, of course, that $H_{t}$ has,
as above, a strict local maximum at $x$ for all $t$. This case,
roughly speaking, corresponds to an SDM $x$ with $d\varphi|_{x}={id}$.

To prove (3.2)‣ for a given $\epsilon>0$, we will use the
standard squeezing argument, i.e., we will bound $H$ from above and
below by two autonomous Hamiltonians $H_{\pm}$ as in Fig. 1 and calculate the Floer homology of $kH_{\pm}$.

Figure 1: The functions $H_{\pm}$

In a Darboux neighborhood $U$ of $x$, the Hamiltonians $H_{\pm}$ are
rotationally symmetric. The Hamiltonian $H_{+}$ is constant and equal
to $c$ near $x$ on a ball of radius $r$ and then sharply decreases
to some $c^{\prime}$ which is close to $c$ and attained on the sphere of
radius $R$. Then, after staying constant on a spherical shell, $H_{+}$
increases to some value $c_{+}$, to accommodate $H$, and becomes
constant. The radii $r<R$ depend on $\epsilon$; namely, we require that
$\pi R^{2}<\epsilon$. The Hamiltonian $H_{-}$ is a bump function decreasing
from its maximum $c$ at $x$ to a large negative value $c_{-}$. Thus we
have

$H_{+}\geq H\geq H_{-}.$

We require $H_{-}$ to have a strict maximum at $x$ with $d^{2}H_{-}(x)=0$.
Then the local Floer homology of $H_{-}^{{\natural}k}$ is equal to ${\mathbb{Z}}_{2}$
and concentrated in degree $n$, i.e., $x$ is also an SDM for $H_{-}$
and all its iterations.

Now, for any $a<b$ outside ${\mathcal{S}}(H^{{\natural}k})$ and ${\mathcal{S}}(H^{{\natural}k}_{\pm})$, we have the maps

induced by monotone homotopies, where $\operatorname{HF}^{(a,\,b)}_{*}\big(H^{{\natural}k}_{\pm}\big)=\operatorname{%
HF}^{(a,\,b)}_{*}\big(kH_{\pm}\big)$ since $H_{\pm}$ are
autonomous Hamiltonians. Therefore, it is sufficient to prove that
the map

To this end, let us assume first that $H_{\pm}$, as above, are
functions on ${\mathbb{R}}^{2n}$ constant outside a neighborhood of $x=0$. The
filtered Floer homology of $H_{\pm}$ is still defined for any interval
$(a,\,b)$ not containing $c_{\pm}$. Moreover, a decreasing homotopy
$H^{s}$ from $H_{+}$ to $H_{-}$ through functions constant outside a
compact set induces a map in Floer homology even when the value of
$H^{s}$ at infinity passes through $(a,\,b)$. Then we have an
isomorphism

when $k$ is sufficiently large and $\delta>0$ is sufficiently small.
Namely, $k$ is so large that $k(c-c^{\prime})>\pi R^{2}$. This is the origin
of the requirement $k>k_{\epsilon}$. Then the homology of $kH_{\pm}$ is
generated by the periodic orbit closest to $x$, and $\delta$ is
chosen so that $kc+\delta$ is smaller than the action of $kH_{-}$ on
this orbit.

The isomorphism (3.4)‣ is established by a straightforward
analysis of periodic orbits and easily follows from the calculation
carried out already in [Ginzburg and Gürel2004]. It is based on two facts:
that

is an isomorphism when $\delta>0$ is small and that
$\operatorname{HF}^{(a,\,b)}_{n+1}\big(kH_{\pm}\big)=0$ for every sufficiently large
interval $(a,\,b)$ containing $kc$ and contained in $(kc_{-},\,kc_{+})$.

It remains to transplant this calculation from ${\mathbb{R}}^{2n}$ to a closed
manifold $M$. The key to this is the fact that the action interval
in question is sufficiently small. This enables one to localize a
calculation of filtered Floer homology by essentially turning action
localization to spatial localization. A general framework for this
process, developed in [Ginzburg and Gürel2009b], is as follows. Let $S$ be a
shell in $M$, i.e., a region between two hypersurfaces and bounding
a contractible domain $V$ in $M$. (To be more precise, $V$ is
bounded in $M$ by a connected component of $\partial S$ and $S\cap V=\emptyset$. The contractibility assumption can be significantly
relaxed.) Furthermore, let $F$ be a Hamiltonian which we require to
be constant on the shell. For any interval $I=(\alpha,\,\beta)$ not
containing $F|_{S}$, consider the subspace of the Floer complex
generated by the orbits of $F$ in $V$ with cappings also contained
in $V$. If necessary, we perturb $F$ in $V$ to make sure that the
orbits with action in $I$ are non-degenerate. Then there exists a
constant $\epsilon(S)>0$ such that, when $|I|=\beta-\alpha<\epsilon(S)$,
this subspace of the Floer complex is actually a subcomplex and,
moreover, a direct summand. [This is an immediate consequence of the
fact that a holomorphic curve crossing $S$ must have energy bounded
away from zero by some constant $\epsilon(S)$.] Furthermore,
continuation maps respect this decomposition as long as the
Hamiltonians remain constant on $S$. (However, the value of the
Hamiltonians on $S$ can enter the interval $I$ during the homotopy.)
Let us denote the resulting Floer homology by $\operatorname{HF}_{*}^{I}(F;V)$.

We apply this construction to $H_{\pm}$ with $S$ being the spherical
shell where $H_{+}=c^{\prime}$ and $V$ being the ball of radius $R$ enclosed
by this shell. (Hence we also need $H_{-}$ to be constant outside
$V$.) As a consequence, (3.4)‣ turns into an isomorphism

entering the map (3.3)‣ as a direct summand. Hence
(3.3)‣ is also non-zero.

The general case where we only have $\|d^{2}H(x)\|<\eta$ is handled in
a similar way, but the construction of $H_{\pm}$ is considerably more
involved and the choice of the modified Hamiltonians $H$ requires
more attention; see [Ginzburg2010] and [Ginzburg and Gürel2009b].
$\square$

Interestingly, no other proof of the Conley conjecture is known for
general symplectic manifolds. A more conceptual or just plain
different argument may shed new light on the nature of the phenomena
considered here and is likely to have other applications. [For the
torus, a different proof is given in the original work [Hingston2009] and
then in [Mazzucchelli2013]. However, it is not clear to us how to
translate that proof to symplectic topological language.]

Remark 3.12.

The part of the proof that does not go through when $M$ is
irrational is the last step, the localization. The difficulty is
that the action spectrum is dense in this case, and necessarily some
of the recappings of degenerate trivial orbits of $F$ in $S$ have
actions in $I$. Thus it is not obvious how to define the Floer
homology localized in $V$. This problem is circumvented in
[Hein2012] by considering the Hamiltonians which have a slight
slope in $S$ rather than being constant. With this modification, the
localization procedure goes through, although the underlying reason
for the localization is now different; see [Hein2012] and [Usher2009].

The collection of all closed symplectic manifolds breaks down into
two classes: those for which the Conley conjecture holds and those
for which the Conley conjecture fails. Of course, the non-trivial
assertion is then that, as we have seen, the former class is
non-empty and even quite large. The situation with closed contact
manifolds is more involved even if we leave aside such fundamental
questions as the Weinstein conjecture and furthermore focus
exclusively on the contact homological properties of the manifold.

First of all, there is a class of contact manifolds
for which every Reeb flow has infinitely many simple closed orbits
because the rank of the contact or symplectic homology grows as a
function of the index or of some other parameter related to the
order of iteration. This phenomenon is studied in, e.g., [Colin and Honda2013],
[Hryniewicz and Macarini2015] and [McLean2012] and the results generalize and are inspired
by a theorem from [Gromoll and Meyer1969], establishing the existence of
infinitely many closed geodesics for manifolds whose free loop space
homology grows. (A technical but important fact closely related to
Theorem3.2 and underpinning the proof is that
the iterates of a given orbit can make only bounded contributions to
the homology; see [Ginzburg and Gürel2010], [Gromoll and Meyer1969], [Hryniewicz and Macarini2015] and [McLean2012]
for various incarnations of this result.) By [Vigué-Poirrier and Sullivan1976], [Abbondandolo and Schwarz2006],
[Salamon and Weber2006] and [Viterbo1999], among contact manifolds in this class are
the unit cotangent bundles $ST^{*}M$ whenever $\pi_{1}(M)=0$ and the
algebra $\operatorname{H}^{*}(M;{\mathbb{Q}})$ is not generated by one element, and some
others; [Colin and Honda2013], [Hryniewicz and Macarini2015] and [McLean2012]. As is already pointed
out in Sect. 2.1, this homologically forced
existence of infinitely many Reeb orbits has very different nature
from the Hamiltonian Conley conjecture where there is no homological
growth.

Then there are contact manifolds admitting Reeb flows with finitely
many closed orbits. Among these are, of course, the standard contact
spheres and, more generally, the pre-quantization circle bundles
over symplectic manifolds admitting torus actions with isolated
fixed points; see Gürel ([Gürel2015], Example 1.13). Note that the class of
such pre-quantization circle bundles includes the Katok–Ziller
flows, i.e., Finsler metrics with finitely many closed geodesics on
$S^{n}$ and on some other manifolds; see [Katok1973] for the original
construction and also [Ziller1983]. Another important group of examples
also containing the standard contact spheres arises from contact
toric manifolds; see [Abreu and Macarini2012]. These two classes (pre-quantization
circle bundles and contact toric manifolds) overlap, but do not
entirely coincide. Although this is not obvious, Reeb flows with
finitely many periodic orbits may have non-trivial dynamics, e.g.,
be ergodic; see [Katok1973].

Finally, as is shown in [Ginzburg et al.2014], there is a non-empty
class of contact manifolds for which every Reeb flow (meeting
certain natural index conditions) has infinitely many simple closed
orbits, although there is no obvious homological growth—the rank
of the relevant contact homology remains bounded. One can expect
this class to be quite large, but at this point such unconditional
existence of infinitely many closed Reeb orbits has only been proved
for the pre-quantization circle bundles of certain aspherical
manifolds; see Theorem4.1. The proof of this theorem is
quite similar to its Hamiltonian counterpart.

This picture is, of course, oversimplified and not even close to
covering the entire range of possibilities, even on the homological
level. For instance, hypothetically, the Reeb flows for overtwisted
contact structures have infinitely many simple closed orbits, but
where should one place such contact structures in our
“classification”? [See [Eliashberg1998] and [Yau2006] for a proof of the existence
of one closed orbit in this case.]

One application of Theorem4.1 is the existence of
infinitely many simple periodic orbits for all low energy levels of
twisted geodesic flows on surfaces of genus $g\geq 2$ with
non-vanishing magnetic field; see Sect. 5.

Just as in the Hamiltonian setting, the mean indices or the actions
and the mean indices of simple periodic orbits of Reeb flows must,
in many instances, satisfy certain resonance relations when the
number of closed orbits is finite. The mean index resonance
relations for the standard contact sphere were discovered by Viterbo
in [Viterbo1989], and the Morse–Bott case for geodesic flows was
considered in [Rademacher1989]. Viterbo’s resonance relations were
generalized to non-degenerate Reeb flows on a broad class of contact
manifolds in [Ginzburg and Kerman2010]. These resonance relations resemble the
equality between two expressions for the Euler characteristic of a
closed manifold: the homological one and the one using indices of
zeroes of a vector field. The role of the homological expression is
now taken by the mean Euler characteristic of the contact homology
of the manifold, introduced in [van Koert2005], and the sum of the indices
is replaced by the sum of certain local invariants of simple closed
orbits. The degenerate case of the generalized Viterbo resonance
relations was studied in [Ginzburg and Gören2015], [Hryniewicz and Macarini2015] and [Long et al.2014] and the Morse–Bott
case in [Espina2014]. There are also variants of resonance relations
involving both the actions and the mean indices; see [Gürel2015]
and also [Ekeland1984] and [Ekeland and Hofer1987].

Leaving aside the exact form of the resonance relations, we only
mention here some of their applications. The first one, in dynamics,
is a contact analog of Theorem2.2: the generic
existence of infinitely many simple closed orbits for a large class
of Reeb flows; see [Ginzburg and Gürel2009c] and also [Ekeland1984], [Rademacher1994] and [Hingston1984] for related earlier results. Another application, also in
dynamics, is to the proof of the existence of at least two simple
closed Reeb orbits on the standard contact $S^{3}$. This result is
further discussed in the next section; see Theorem4.3. (We
refer the reader to [Gürel2015] for some other applications in
dynamics.) Finally, on the topological side, the resonance relations
can be used to calculate the mean Euler characteristic, [Espina2014].

Consider a closed symplectic manifold $(M,\omega)$ such that the
form $\omega$ or, to be more precise, its cohomology class
$[\omega]$ is integral, i.e., $[\omega]\in\operatorname{H}^{2}(M;{\mathbb{Z}})/\operatorname{Tor}$.
Let $\pi\colon P\to M$ be an $S^{1}$-bundle over $M$ with first Chern
class $-[\omega]$. The bundle $P$ admits an $S^{1}$-invariant 1-form
$\alpha_{0}$ such that $d\alpha_{0}=\pi^{*}\omega$ and $\alpha_{0}(R_{0})=1$,
where $R_{0}$ is the vector field generating the $S^{1}$-action on $P$.
In other words, when we set $S^{1}={\mathbb{R}}/{\mathbb{Z}}$ and identify the Lie algebra
of $S^{1}$ with ${\mathbb{R}}$, the form $\alpha_{0}$ is a connection form on $P$
with curvature $\omega$. (Note our sign convention.)

Clearly, $\alpha_{0}$ is a contact form with Reeb vector field $R_{0}$,
and the connection distribution $\xi=\ker\alpha_{0}$ is a contact
structure on $P$. Up to a gauge transformation, $\xi$ is independent
of the choice of $\alpha_{0}$. The circle bundle $P$ equipped with
this contact structure is usually referred to as a
pre-quantization circle bundle or a Boothby–Wang bundle.
Also, recall that a degree two (real) cohomology class on $P$ is
said to be atoroidal if its integral over any smooth map
${\mathbb{T}}^{2}\to P$ is zero. (Such a class is necessarily aspherical.)
Finally, in what follows, we will denote by ${\mathfrak{f}}$ the free homotopy
class of the fiber of $\pi$.

The main tool used in the proof of Theorem4.1 stated
below is the cylindrical contact homology. As is well known, to have
this homology defined for a contact form $\alpha$ on any closed
contact manifold $P$ one has to impose certain additional
requirements on the closed Reeb orbits of $\alpha$. (See [Bourgeois2009]; [Eliashberg et al.2000] and references therein for the definition and a detailed
discussion of contact homology.) Namely, following [Ginzburg et al.2014], we
say that a non-degenerate contact form $\alpha$ is
index–admissible if its Reeb flow has no contractible closed
orbits with Conley–Zehnder index $2-n$ or $2-n\pm 1$, where $\dim P=2n+1$. In general, $\alpha$ or its Reeb flow is index–admissible
when there exists a sequence of non-degenerate index–admissible
forms $C^{1}$-converging to $\alpha$.

This requirement is usually satisfied when $(P,\alpha)$ has some
geometrical convexity properties. For instance, the Reeb flow on a
strictly convex hypersurface in ${\mathbb{R}}^{2m}$ is index–admissible,
[Hofer et al.1998]. Likewise, as is observed in [Benedetti2014], the
twisted geodesic flow on a low energy level for a symplectic
magnetic field on a surface of genus $g\geq 2$ is index–admissible;
see Sect. 5 for more details. Finally, let us
call a closed Reeb orbit $x$ non-degenerate (or weakly
non-degenerate, SDM, etc.) if its Poincaré return map is
non-degenerate (or, respectively, weakly non-degenerate, SDM, etc.),
cf. Sect. 2.2.1. (The Poincaré return map is
the map, or rather the germ of a map, $\Sigma\to\Sigma$ defined on
a small cross section at $x(0)$ and sending a point $z\in\Sigma$ to
the first intersection of the Reeb orbit through $z$ with $\Sigma$.)

Theorem 4.1.

(Contact Conley conjecture, [Ginzburg et al.2014]) Assume that

(i)

$M$
is aspherical, i.e.,
$\pi_{r}(M)=0$
for all
$r\geq 2$
,
and

(ii)

$c_{1}(\xi)\in\operatorname{H}^{2}(P;{\mathbb{R}})$
is atoroidal.

Let $\alpha$ be an index–admissible contact form on the
pre-quantization bundle $P$ over $M$, supporting $\xi$. Then the
Reeb flow of $\alpha$ has infinitely many simple closed orbits with
contractible projections to $M$. Assume furthermore that the Reeb
flow has finitely many periodic orbits in the free homotopy class
${\mathfrak{f}}$ of the fiber and that these orbits are weakly non-degenerate.
Then for every sufficiently large prime $k$ the Reeb flow of
$\alpha$ has a simple closed orbit in the class ${\mathfrak{f}}^{k}$, and all
classes ${\mathfrak{f}}^{k}$ are distinct.

It follows from Theorem4.1 and the discussion below that,
when the Reeb flow of $\alpha$ is weakly non-degenerate, the number
of simple periodic orbits of the Reeb flow of $\alpha$ with period
(or equivalently action) less than $a>0$ is bounded from below by
$C_{0}\cdot a/\ln a-C_{1}$, where $C_{0}=\inf\alpha(R_{0})$ and $C_{1}$
depends only on $\alpha$. As mentioned in Sect.
2.2, this is a typical lower growth bound in the
Conley conjecture type results. Note also that the weak
non-degeneracy requirement here plays a technical role and probably
can be eliminated.

The key to the proof of Theorem4.1 is the observation
that, as a consequence of (i), all free homotopy classes ${\mathfrak{f}}^{k}$,
$k\in{\mathbb{N}}$, are distinct and hence give rise to an ${\mathbb{N}}$-grading of the
cylindrical contact homology of $(P,\alpha)$. (In fact, it would be
sufficient to assume that $[\omega]$ is aspherical and $\pi_{1}(M)$ is
torsion free; both of these requirements follow from (i).) This
grading plays essentially the same role as the order of iteration in
the Hamiltonian Conley conjecture. With this observation in mind,
the proof of the weakly non-degenerate case is quite similar to its
Hamiltonian counterpart. [Condition (ii) is purely technical and
most likely can be dropped.]

To complete the proof, one then has to deal with the case where the
Reeb flow of $\alpha$ has a simple SDM orbit, i.e., a simple
isolated orbit with an SDM Poincaré return map. This is also done
similarly to the Hamiltonian case, but there are some nuances.

Consider a closed contact manifold $(P^{2n+1},\ker\alpha)$ with a
strong symplectic filling $(W,\omega)$, i.e., $W$ is a compact
symplectic manifold such that $P=\partial W$ with $\omega|_{P}=d\alpha$ and
a natural orientation compatibility condition is satisfied. Let
${\mathfrak{c}}$ be a free homotopy class of loops in $W$.

Theorem 4.2.

([Ginzburg et al.2013]; [Ginzburg et al.2014]) Assume that the Reeb flow of $\alpha$ has a
simple closed SDM orbit in the class ${\mathfrak{c}}$ and one of the following
requirements is met:

•

$W$
is symplectically aspherical and
${\mathfrak{c}}=1$
, or

•

$\omega$
is exact and
$c_{1}(TW)=0$
in
$H^{2}(W;{\mathbb{Z}})$
.

Then the Reeb flow of $\alpha$ has infinitely many simple periodic
orbits.

This result is a contact analog of Theorem3.8.
Theorem4.1 readily follows from the first case of Theorem4.2 where we take the pre-quantization disk bundle
over $M$ as $W$. [Here we only point out that $\pi_{2}(W)=\pi_{2}(M)=0$
since $M$ is aspherical and refer the reader to [Ginzburg et al.2014] for more
details.]

The proof of Theorem4.2 uses the filtered
linearized contact homology. To be more specific, denote by
$\operatorname{HC}_{*}^{(a,\,b)}(\alpha;W,{\mathfrak{c}}^{k})$ the linearized contact homology
of $(P,\alpha)$ with respect to the filling $(W,\omega)$ for the
action interval $(a,\,b)$ and the free homotopy class ${\mathfrak{c}}^{k}$,
graded by the Conley–Zehnder index. Set $\Delta=\Delta(x)$ and
$c={\mathcal{A}}(x)$ where $x$ is the SDM orbit from the theorem. Similarly to
the Hamiltonian case (cf. Theorem3.9), one first shows
that, under the hypotheses of the theorem, for any $\epsilon>0$ there
exists $k_{\epsilon}\in{\mathbb{N}}$ such that

$\operatorname{HC}_{k\Delta+n+1}^{(kc+\delta_{k},\,kc+\epsilon)}(\alpha;W,{%
\mathfrak{c}}^{k})\neq 0\textrm{ for all }k>k_{\epsilon}\textrm{ and some }%
\delta_{k}<\epsilon.$

(4.1)

Theorem4.2 is a consequence of (4.1)‣
(see [Ginzburg et al.2014]), although the argument is less obvious than its
Hamiltonian counterpart for, say, symplectic CY manifolds.

The proof of (4.1)‣ given in [Ginzburg et al.2013] follows the same
path as the proof of Theorem3.9. Namely, we squeeze the
form $\alpha$ between two contact forms $\alpha_{\pm}$ constructed
using the Hamiltonians $H_{\pm}$ near the SDM orbit, calculate the
relevant contact homology for $\alpha_{\pm}$ (or rather a direct
summand in it), and show that the map in contact homology induced by
the cobordism from $\alpha_{+}$ to $\alpha_{-}$ is non-zero. This map
factors through
$\operatorname{HC}_{k\Delta+n+1}^{(kc+\delta_{k},\,kc+\epsilon)}(\alpha;W,{%
\mathfrak{c}}^{k})$, and
hence this group is also non-trivial.

Note that Theorem4.2 as stated, without further
assumptions on ${\mathfrak{c}}$, affords no control on the free homotopy
classes of the simple orbits or their growth rate. A related point
is that, at the time of this writing, there seems to be no
satisfactory version of Theorem4.2 which would
not rely on the existence of the filling $W$. The difficulty is that
without a filling one is forced to work with cylindrical contact
homology to prove a variant of (4.1)‣, but then it is not
clear if the forms $\alpha_{\pm}$ can be made index–admissible
without additional assumptions on $\alpha$ along the lines of
index–positivity. Such a filling–free version of the theorem
would, for instance, enable one to eliminate the weak non-degeneracy
assumption in the growth assertion in Theorem4.1. Another
serious limitation of Theorem4.2 is that the SDM
orbit is required to be simple. This condition, which is quite
restrictive but probably purely technical, is used in the proof in a
crucial way to construct the forms $\alpha_{\pm}$.

Another application of Theorem4.2 considered in
[Ginzburg et al.2013] (and also in [Ginzburg and Gören2015]; [Liu and Long2013]) is the following result
originally proved in [Cristofaro-Gardiner and Hutchings2012].

Theorem 4.3.

The Reeb flow of a contact form $\alpha$ supporting the standard
contact structure on $S^{3}$ has at least two simple closed orbits.

In fact, a much stronger result holds. Namely, every Reeb flow on a
closed three-manifold has at least two simple closed Reeb orbits.
This fact is proved in [Cristofaro-Gardiner and Hutchings2012] using the machinery of embedded
contact homology and is outside the scope of this survey. The idea
of the proof from [Ginzburg et al.2013] is that if a Reeb flow on the standard
contact $S^{3}$ had only one simple closed orbit $x$, this orbit would
be an SDM, and, by Theorem4.2, the flow would
have infinitely many periodic orbits. Showing that $x$ is indeed an
SDM requires a rather straightforward index analysis with one
non-trivial ingredient used to rule out a certain index pattern. In
[Ginzburg et al.2013], this ingredient is strictly three-dimensional and comes
from the theory of finite energy foliations (see [Hofer et al.1995]; [Hofer et al.1996]).
The argument in [Ginzburg and Gören2015]; [Liu and Long2013] uses a variant of the resonance
relation for degenerate Reeb flows proved in [Ginzburg and Gören2015] and [Long et al.2014].
Theorem4.2 can also be applied to give a simple
proof, based on the same idea, of the result from [Bangert and Long2010] that any
Finsler geodesic flow on $S^{2}$ has at least two closed geodesics;
see [Ginzburg and Gören2015]. [Of course, this fact also immediately follows from
[Cristofaro-Gardiner and Hutchings2012].]

Interestingly, no multiplicity results along the lines of
Theorem4.3 have been proved in higher dimensions without
restrictive additional assumptions on the contact form.
Conjecturally, every Reeb flow on the standard contact sphere
$S^{2n-1}$ has at least $n$ simple closed Reeb orbits. This
conjecture has been proved when the contact form comes from a
strictly convex hypersurface in ${\mathbb{R}}^{2n}$ and the flow is
non-degenerate or $2n\leq 8$; see [Long and Zhu2002], [Long2002] and [Wang2013] and references
therein. In the degenerate strictly convex case, the lower bound is
$\lfloor n/2\rfloor+1$. Without any form of a convexity
assumption, it is not even known if a Reeb flow on the standard
contact $S^{5}$ must have at least two simple closed orbits. It is
easy to see, however, that a non-degenerate Reeb flow on the
standard $S^{2n-1}$ has at least two simple closed orbits; see,
e.g., Gürel ([Gürel2015], Rmk. 3.3).

We conclude this section by pointing out that the machinery of
contact homology which the proof of Theorem4.1 relies on
is yet to be fully put on a rigorous basis.

The results from Sect. 4 have immediate applications
to the dynamics of twisted geodesic flows. These flows give a
Hamiltonian description of the motion of a charge in a magnetic
field on a Riemannian manifold.

To be more precise, consider a closed Riemannian manifold $M$ and
let $\sigma$ be a closed 2-form (a magnetic field) on $M$.
Let us equip $T^{*}M$ with the twisted symplectic structure
$\omega=\omega_{0}+\pi^{*}\sigma$, where $\omega_{0}$ is the standard
symplectic form on $T^{*}M$ and $\pi\colon T^{*}M\to M$ is the natural
projection, and let $K$ be the standard kinetic energy Hamiltonian
on $T^{*}M$ arising from the Riemannian metric on $M$. The Hamiltonian
flow of $K$ on $T^{*}M$ governs the motion of a charge on $M$ in the
magnetic field $\sigma$ and is referred to as a twisted
geodesic or magnetic flow. In contrast with the geodesic
flow (the case $\sigma=0$), the dynamics of a twisted geodesic flow
on an energy level depends on the level. In particular, when $M$ is
a surface of genus $g\geq 2$, the example of the horocycle flow
shows that a symplectic magnetic flow need not have periodic orbits
on all energy levels. Furthermore, the dynamics of a twisted
geodesic flow also crucially depends on whether one considers low or
high energy levels, and the methods used to study this dynamics
further depend on the specific properties of $\sigma$, i.e., on
whether $\sigma$ is assumed to be exact or symplectic or, when $M$
is a surface, non-exact but changing sign, etc.

The existence problem for periodic orbits of a charge in a magnetic
field was first addressed in the context of symplectic geometry by
V.I. Arnold in the early 80s; see [Arnold1986]; [Arnold1988]. Namely, Arnold
proved that, as a consequence of the Conley–Zehnder theorem,
[Conley and Zehnder1983a], a twisted geodesic flow on ${\mathbb{T}}^{2}$ with symplectic
magnetic field has periodic orbits on all energy levels when the
metric is flat and on all low energy levels for an arbitrary metric,
[Arnold1988]. It is still unknown if the latter result can be
extended to all energy levels; however it was generalized to all
surfaces in [Ginzburg1987].

Example 5.1.

Assume that $M$ is a surface and let $\sigma=f\,dA$, where $dA$ is
an area form. Assume furthermore that $f$ has a non-degenerate
critical point at $x$. Then it is not hard to see that essentially
by the inverse function theorem the twisted geodesic flow on a low
energy level has a closed orbit near the fiber over $x$.

Since Arnold’s work, the problem has been studied in a variety of
settings. We refer the reader to, e.g., [Ginzburg1994] for more
details and references prior to 1996 and to, e.g.,
[Asselle and Benedetti2014], [Abbondandolo et al.2014], [Contreras et al.2004], [Ginzburg et al.2014] and [Kerman1999] for some more recent
results and references.

Here we focus exclusively on the case where the magnetic field form
$\sigma$ is symplectic (i.e., non-vanishing when $\dim M=2$), and we
are interested in the question of the existence of periodic orbits
on low energy levels. In this setting, in all dimensions, the
existence of at least one closed orbit with contractible projection
to $M$ was proved in [Ginzburg and Gürel2009a] and [Usher2009].

Furthermore, when $\sigma$ is symplectic, we can also think of $M$
as a symplectic submanifold of $(T^{*}M,\omega)$ and $K$ as a
Hamiltonian on $T^{*}M$ attaining a Morse–Bott non-degenerate local
minimum $K=0$ at $M$. Thus we can treat the problem of the existence
of periodic orbits on a low energy level $P_{\epsilon}=\{K=\epsilon\}$ as a
generalization of the classical Moser–Weinstein theorem (see
[Moser1976]; [Weinstein1973]), where an isolated non-degenerate minimum is
replaced by a Morse–Bott non-degenerate minimum and the critical
set is symplectic. This is the point of view taken in [Kerman1999]
and then in, e.g., [Ginzburg and Gürel2004]; [Ginzburg and Gürel2009a]. To prove the existence
of a periodic orbit on every low energy level one first shows that
almost all low energy levels carry a periodic orbit with mean index
in a certain range depending only on $\dim M$ and having
contractible projection to $M$; see, e.g., [Ginzburg and Gürel2004] and [Schlenk2006].
This fact does not really require $M$ to be symplectic; it is
sufficient to assume that $\sigma\neq 0$. Then a Sturm theory type
argument is used in [Ginzburg and Gürel2009a] and [Usher2009] to show that long orbits
must necessarily have high index, and hence, by the Arzela–Ascoli
theorem, every low energy level carries a periodic orbit. At this
step, the assumption that the Hessian $d^{2}K$ is positive definite on
the normal bundle to $M$ becomes essential, cf. Ginzburg and Gürel ([Ginzburg and Gürel2004], Sect.
2.4).

There are also multiplicity results. Already in [Arnold1986]; [Arnold1988],
it was proved that when $M={\mathbb{T}}^{2}$ and $\sigma$ is symplectic, there
are at least three (or four in the non-degenerate case) periodic
orbits on every low energy level $P_{\epsilon}$. Furthermore, Arnold also
conjectured that the lower bounds on the number of periodic orbits
are governed by Morse theory and Lusternik–Schnirelmann theory as
in the Arnold conjecture whenever $\sigma$ is symplectic and
$\epsilon>0$ is small enough. These lower bounds were then proved for
surfaces in [Ginzburg1987].

For the torus the proof is particularly simple. Let us fix a flat
connection on $P_{\epsilon}={\mathbb{T}}^{2}\times S^{1}$. When $\epsilon>0$ is small, the
horizontal sections are transverse to $X_{K}$, and one can show that
the resulting Poincaré return map is a Hamiltonian diffeomorphism
${\mathbb{T}}^{2}\to{\mathbb{T}}^{2}$; see, e.g., [Ginzburg1987] and [Levi2003]. Now it remains to apply
the Conley–Zehnder theorem. Note that this argument captures only
the short orbits, i.e., the orbits in the homotopy class of the
fiber. Likewise, the proof for other surfaces in [Ginzburg1987]
captures only the orbits that stay close to a fiber and wind around
it exactly once. In higher dimensions, however, it is not entirely
clear how to define such short orbits. The difficulty arises from
the fact that $d^{2}K$ has several “modes” in every fiber, and the
modes can vary significantly and bifurcate from one fiber to
another. Furthermore, the Weinstein–Moser theorem provides a
hypothetical lower bound which is different from the one coming from
the Arnold conjecture perspective; see [Kerman1999]. Without a
distinguished class of short orbits to work with, one is forced to
consider all periodic orbits and, already for $M={\mathbb{T}}^{2}$, use the
Conley conjecture type results in place of the Arnold conjecture.
Hypothetically, as is observed in [Ginzburg and Gürel2009a], every low energy
level should carry infinitely many simple periodic orbits, at least
when $(M,\sigma)$ is a symplectic CY manifold. This is still a
conjecture when $\dim M>2$, but in dimension two the question has
been recently settled in [Ginzburg et al.2014]. Namely, we have

Theorem 5.2.

([Ginzburg et al.2014]) Assume that $M$ is a surface of genus $g\geq 1$
and $\sigma$ is symplectic. Then for every small $\epsilon>0$, the flow
of $K$ has infinitely many simple periodic orbits on $P_{\epsilon}$ with
contractible projections to $M$. Moreover, assume that the flow has
finitely many periodic orbits in the free homotopy class ${\mathfrak{f}}$ of
the fiber. Then for every sufficiently large prime $k$ there is a
simple periodic orbit in the class ${\mathfrak{f}}^{k}$, and all such classes are
distinct.

When $M={\mathbb{T}}^{2}$, the theorem immediately follows from Arnold’s cross
section argument once one uses the Conley conjecture for ${\mathbb{T}}^{2}$
(proved in [Franks and Handel2003]) instead of the Conley–Zehnder theorem; see
[Ginzburg and Gürel2009a]. When $g\geq 2$, Theorem5.2 (almost)
follows from Theorem4.1 since $P_{\epsilon}$ has contact type
and the flow is index–admissible as observed in [Benedetti2014]. The part
that is not a consequence of Theorem4.1 is the existence
of a simple periodic orbit in the class ${\mathfrak{f}}^{k}$ for a large prime
$k$ without any non-degeneracy assumptions. This is proved by
applying the second case of Theorem4.2 to the
disjoint union $P_{\epsilon}\sqcup P_{E}$, where $E$ is large, with the
filling $W$ formed by the part of $T^{*}M$ between these two energy
levels, and ${\mathfrak{c}}={\mathfrak{f}}$. The proof of Theorem5.2
heavily relies on the machinery of cylindrical contact homology via
its dependence on Theorem4.1. Note, however, that in the
present setting one might be able to circumvent foundational
difficulties by using automatic transversality results
from [Hutchings and Nelson2014]. Alternatively, one could work with the linearized
contact homology or the equivariant symplectic homology for the
filling $W$, entirely avoiding foundational problems in the latter
case.

Two difficulties arise in extending Theorem5.2 to
higher dimensions. One is that the energy levels do not have contact
type, and hence the standard contact or symplectic homology
techniques are not applicable. This difficulty seems to be more
technical than conceptual: using Sturm theory as in [Ginzburg and Gürel2009a] one
can still associate to a level a variant of symplectic homology
generated by periodic orbits on the level. A more serious obstacle
is the lack of filtration by the free homotopy classes ${\mathfrak{f}}^{k}$,
which plays a central role in the proof.

There seems to be no reason to expect Theorem5.2 to
hold for $S^{2}$. However, no counterexamples are known. For instance,
let us consider the round metric on $S^{2}$ and a non-vanishing
magnetic field $\sigma$ symmetric with respect to rotations about
the $z$ axis. The twisted geodesic flow on every energy level is
completely integrable. It would be useful and illuminating to
analyze this flow and check if it has infinitely many periodic
orbits on every (low or high) energy level.

It is conceivable that for any magnetic field, every sufficiently
low energy level carries infinitely many periodic orbits. For exact
magnetic fields on closed surfaces this is proved for almost all low
energy levels in [Abbondandolo et al.2014] by methods from the “classical
calculus of variations”; see, e.g., [Asselle and Benedetti2015] for related results
and further references. It would be extremely interesting to
understand this phenomenon of “almost existence of infinitely many
periodic orbits” from a symplectic topology perspective and
generalize it to higher dimensions. Furthermore, even in dimension
two, no examples of magnetic flows with finitely many periodic
orbits on arbitrarily low energy levels are known. For instance, it
is not known if the completely integrable twisted geodesic flow on
$S^{2}$ with an exact $S^{1}$-invariant magnetic field $\sigma$ has
infinitely many periodic orbits on only almost all low energy levels
or in fact on all such levels. (Note that the Katok–Ziller flows
from [Katok1973]; [Ziller1983] correspond to high energy levels.)

Even when the Conley conjecture fails, the existence of infinitely
many simple periodic orbits is, as we have already seen, a generic
feature of Hamiltonian diffeomorphisms (and Reeb flows) for a broad
class of manifolds. There is, however, a different and more
interesting, from our perspective, phenomenon responsible for the
existence of infinitely many periodic orbits. The starting point
here is a celebrated theorem of Franks.

Theorem 6.1.

([Franks1992]; [Franks1996]) Any Hamiltonian diffeomorphism $\varphi$ of
$S^{2}$ with at least three fixed points has infinitely many simple
periodic orbits.

In fact, the theorem, already in its original form, was proved for
area preserving homeomorphisms. This aspect of the problem is
outside the scope of the paper, and here we focus entirely on smooth
maps. Furthermore, in the setting of the theorem, there are also
strong growth results; see [Franks and Handel2003], [Le Calvez2006] and [Kerman2012] for this and other
refinements of Theorem6.1. The proof of the theorem given
in [Franks1992]; [Franks1996] utilized methods from low–dimensional dynamics.
Recently, a purely symplectic topological proof of the theorem was
obtained in [Collier et al.2012]; see also [Bramham and Hofer2012] for a different
approach.

Outline of the proof from[Collier et al.2012] Arguing by contradiction and passing if necessary to an iteration of
$\varphi$, we can assume that $\varphi$ has finitely many periodic
points, that all these points are fixed points and that there are at
least three fixed points. Applying a variant of the resonance
relations from [Ginzburg and Kerman2010] combined with Theorem3.8 and a simple topological argument, it is not
hard to see that there must be (at least) two fixed points $x$ and
$y$ with irrational mean indices and at least one point $z$ with
zero mean index. Note that, since $\dim S^{2}=2$, the points $x$ and
$y$ are elliptic and strongly non-degenerate, and $z$ is either
degenerate or hyperbolic.

In the former case, we glue together two copies of $S^{2}$ punctured
at $y$ and $z$ by inserting narrow cylinders at the seams as in
Arnold ([Arnold1989], App. 9). As a result, we obtain a torus ${\mathbb{T}}^{2}$, and the
Hamiltonian diffeomorphism $\varphi$ gives rise to an area
preserving map $\psi\colon{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}$. This map is not necessarily
a Hamiltonian diffeomorphism, but it is symplectically isotopic to
${id}$ and its Floer homology $\operatorname{HF}_{*}(\psi)$ is defined. Hence, either
$\operatorname{HF}_{*}(\psi)=0$ or $\operatorname{HF}_{*}(\psi)\cong\operatorname{H}_{*+1}({\mathbb{T}}^{2})$ when $\psi$ is
Hamiltonian. Now one shows that, roughly speaking, any of the points
$x^{\pm}\in{\mathbb{T}}^{2}$ corresponding to $x$ represents a non-trivial
homology class of degree different from 0 and $\pm 1$, which is
impossible.

When $z$ is a hyperbolic point, we use the points $x$ and $y$ to
produce the torus, and again a simple Floer homological argument
leads to a contradiction. Indeed, for a sufficiently large iteration
of $\psi$, each elliptic point has large Conley–Zehnder index,
since Theorem3.8 rules out SDM points, and each
hyperbolic point has even index. Moreover, hyperbolic points give
rise to non-trivial homology classes (cf. [Ginzburg and Gürel2009c], Thm. 1.7). Thus $\operatorname{HF}_{*}(\psi)\neq 0$ but $\operatorname{HF}_{1}(\psi)=0$,
which is again impossible. (Alternatively, one can just apply
Theorem6.2 below to deal with this case.)
$\square$

Even though all proofs of Franks’ theorem are purely
low-dimensional, it is tempting to think of the result as a
particular case of a more general phenomenon. For instance, one
hypothetical generalization of Franks’ theorem would be that a
Hamiltonian diffeomorphism with more than necessary
non-degenerate (or just homologically non-trivial in the sense of
Sect. 3.1.2) fixed points has infinitely many periodic
orbits. Here more than necessary is usually interpreted as a
lower bound arising from some version of the Arnold conjecture. For
${\mathbb{CP}}^{n}$, the expected threshold is $n+1$ and, in particular, it is
2 for $S^{2}$ as in Franks’ theorem, cf. Hofer and Zehnder ([Hofer and Zehnder1994], p. 263).

However, this conjectural generalization of Franks’ theorem seems to
be too restrictive, and from the authors’ perspective it is fruitful
to put the conjecture in a broader context. Namely, it appears that
the presence of a fixed point that is unnecessary from a homological
or geometrical perspective is already sufficient to force the
existence of infinitely many simple periodic orbits. Let us now
state some recent results in this direction.

Theorem 6.2.

([Ginzburg and Gürel2014]) A Hamiltonian diffeomorphism of ${\mathbb{CP}}^{n}$
with a hyperbolic periodic orbit has infinitely many simple periodic
orbits.

Here, clearly, the hyperbolic periodic orbit is unnecessary from
every perspective. In contrast with Franks’ theorem and the Conley
conjecture type results, at the time of writing, there are no growth
results in this setting. The theorem actually holds for a broader
class of manifolds $M$, and the requirements on $M$ can be stated
purely in terms of the quantum homology of $M$; see Ginzburg and Gürel ([Ginzburg and Gürel2014], Thm. 1.1). Among the manifolds meeting these requirements
are, in addition to ${\mathbb{CP}}^{n}$, the complex Grassmannians $\operatorname{Gr}(2;N)$,
$\operatorname{Gr}(3;6)$ and $\operatorname{Gr}(3;7)$; the products ${\mathbb{CP}}^{m}\times P^{2d}$
and $\operatorname{Gr}(2;N)\times P^{2d}$, where $P$ is symplectically aspherical
and $d\leq m$ in the former case and $d\leq 2$ in the latter; and
the monotone products ${\mathbb{CP}}^{m}\times{\mathbb{CP}}^{m}$. There is also a variant
of the theorem for non-contractible hyperbolic orbits, which is
applicable to, for example, the product ${\mathbb{CP}}^{m}\times P^{2d}$. Note
also that the generalization of Franks’ theorem to ${\mathbb{CP}}^{n}$, at least
for non-degenerate Hamiltonian diffeomorphisms, would follow if one
could replace in Theorem6.2 a hyperbolic fixed
point by a non-elliptic one.

Another result fitting into this context is the following.

Theorem 6.3.

([Gürel2014]) Let $\varphi\colon{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n}$ be a
Hamiltonian diffeomorphism generated by a Hamiltonian equal to a
hyperbolic quadratic form $Q$ at infinity (i.e., outside a compact
set) such that $Q$ has only real eigenvalues. Assume that $\varphi$
has finitely many fixed points, and one of these points, $x$, is
non-degenerate (or just isolated and homologically non-trivial) and
has non-zero mean index. Then $\varphi$ has simple periodic orbits
of arbitrarily large period.

As a consequence, regardless of whether the fixed-point set is
finite or not, $\varphi$ has infinitely many simple periodic orbits.
In this theorem the condition that the eigenvalues of $Q$ are real
can be slightly relaxed. Conjecturally, it should be enough to
require $Q$ to be non-degenerate and $x$ to have mean index
different from the mean index of the origin for $Q$. However,
hyperbolicity of $Q$ is used in an essential way in the proof of the
theorem. Also, interestingly, in contrast with Franks’ theorem, the
requirement that $x$ is homologically non-trivial is essential and
cannot be omitted, even in dimension two. As an easy consequence of
Theorem6.3, we obtain

Theorem 6.4.

([Gürel2014]) Let $\varphi\colon{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n}$, where
$2n=2$ or $2n=4$, be a Hamiltonian diffeomorphism generated by a
Hamiltonian equal to a hyperbolic quadratic form $Q$ at infinity as
in Theorem6.3. Assume that $\varphi$ is strongly
non-degenerate and has at least two (but finitely many) fixed
points. Then $\varphi$ has simple periodic orbits of arbitrarily
large period.

In the two-dimensional case, a stronger result is proved in
Abbondandolo ([Abbondandolo2001], Thm. 5.1.9). In the setting of
Theorem6.3 and Theorem6.4, one can be more precise about which simple periods
occur. Namely, for a certain integer $m>0$, starting with a
sufficiently large prime number, among every $m$ consecutive primes,
there exists at least one prime which is the period of a simple
periodic orbit. Thus, as in many other results of this type, we have
the lower growth bound ${const}\cdot k/\ln k$.

Theorem6.2 and, with some extra work, Theorem6.3 imply the case of Franks’ theorem where $\varphi$ is
assumed to have a hyperbolic periodic orbit, e.g., when $\varphi$ is
non-degenerate. Furthermore, it is conceivable that one could prove
Franks’ theorem as a consequence of Theorem6.2.
Such a proof would certainly be of interest, but it would most
likely be much more involved than the argument in [Collier et al.2012].

Let us now turn to non-contractible orbits. Recall
that for a (time-dependent) Hamiltonian flow $\varphi_{H}^{t}$ generated
by a Hamiltonian $H\colon S^{1}\times M\to{\mathbb{R}}$ there is a one-to-one
correspondence between the one-periodic orbits of $\varphi_{H}^{t}$ and
the fixed points of $\varphi=\varphi_{H}$. Furthermore, as is easy to
see from the proof of the Arnold conjecture, the free homotopy class
of an orbit $x$ is independent of the Hamiltonian generating the
time-one map $\varphi$. Thus the notion of a contractible
one-periodic orbit (or even a “contractible fixed point”) of
$\varphi$ is well-defined. Of course, the same applies to
$k$-periodic orbits.

On a closed symplectic manifold $M$ a Hamiltonian diffeomorphism
need not have non-contractible one-periodic orbits. Indeed, the
Hamiltonian Floer homology vanishes for any non-trivial free
homotopy class when $M$ is compact, since all one-periodic orbits of
a $C^{2}$-small autonomous Hamiltonian are its critical points (hence
contractible). Thus, from a homological perspective,
non-contractible periodic orbits are totally unnecessary.

To state our next result, recall that a symplectic form $\omega$ on
$M$ is said to be atoroidal if for every map $v\colon{\mathbb{T}}^{2}\to M$,
the integral of $\omega$ over $v$ vanishes. We have

Theorem 6.5.

([Gürel2013]) Let $M$ be a closed symplectic manifold
equipped with an atoroidal symplectic form $\omega$. Assume that a
Hamiltonian diffeomorphism $\varphi$ of $M$ has a non-degenerate
one-periodic orbit $x$ with homology class $[x]\neq 0$ in
$\operatorname{H}_{1}(M;{\mathbb{Z}})/\operatorname{Tor}$ and that the set of one-periodic orbits in the
class $[x]$ is finite. Then, for every sufficiently large prime $p$,
the Hamiltonian diffeomorphism $\varphi$ has a simple periodic orbit
in the homology class $p[x]$ and with period either $p$ or $p^{\prime}$,
where $p^{\prime}$ is the first prime greater than $p$.

Thus the number of simple non-contractible periodic orbits with
period less than or equal to $k$, or the number of distinct homology
classes represented by such orbits, is bounded from below by ${const}\cdot k/\ln k$. An immediate consequence of the theorem is that
$\varphi$ has infinitely many simple periodic orbits with homology
classes in ${\mathbb{N}}[x]$ whether or not the set of one-periodic orbits (in
the class $[x]$) is finite. Moreover, in this theorem, as in Theorem6.3, the non-degeneracy condition can be relaxed and
replaced by the much weaker requirement that $x$ is isolated and
homologically non-trivial. Finally, in both theorems, the orbit $x$
need not be one-periodic; the theorems (with obvious modifications)
still hold when $x$ is just a periodic orbit.

Among the manifolds meeting the requirements of Theorem6.5
are, for instance, closed Kähler manifolds with negative sectional
curvature and, more generally, any closed symplectic manifold with
$[\omega]|_{\pi_{2}(M)}=0$ and hyperbolic $\pi_{1}(M)$. Furthermore,
Hamiltonian diffeomorphisms having a periodic orbit in a non-trivial
homology class exist in abundance. It is plausible that a
$C^{1}$-generic, or even $C^{\infty}$-generic, Hamiltonian
diffeomorphism has an orbit in a non-trivial homology class when the
fundamental group (or the first homology group) of $M$ is large
enough; see [Tal2013] for some possibly relevant results for
surfaces. However, as is easy to see, already for $M={\mathbb{T}}^{2}$, a fixed
Hamiltonian diffeomorphism need not have non-contractible periodic
orbits (e.g., $\varphi_{H}$ for a small bump function $H$), and even
$C^{\infty}$-generically one cannot prescribe the homology class of an
orbit in advance; [Ginzburg and Gürel2015] and [Gürel2013].

Hypothetically, one can expect an analog of the theorem to hold when
the condition that $\omega$ is atoroidal is omitted or relaxed,
e.g., replaced by the requirement that $(M,\omega)$ is toroidally
monotone. We refer the reader to [Ginzburg and Gürel2015] for some further
results in this direction.

The proofs of all these theorems are based on the same idea that an
unnecessary periodic orbit is a seed creating infinitely many
periodic orbits. In Theorem6.3 and Theorem6.5 the
argument is that, roughly speaking, the change in filtered Floer
homology, for a carefully chosen action range (and/or degree),
between different iterations of $\varphi$ requires new simple
periodic orbits to be created. The proof of Theorem6.2 relies on a result, perhaps of independent
interest, asserting that the energy needed for a Floer connecting
trajectory of an iterated Hamiltonian to approach a hyperbolic orbit
and cross its fixed neighborhood cannot become arbitrarily small: it
is bounded away from zero by a constant independent of the order of
iteration. Then the product structure in quantum homology is used to
show that there must be Floer connecting trajectories with energy
converging to zero for some sequence of iterations unless new
periodic orbits are created.

Just like Hamiltonian diffeomorphisms, Reeb flows
with “unnecessary” periodic orbits can be expected to have
infinitely many simple periodic orbits. However, at the time of this
writing, there is little evidence supporting this conjecture, and
all the relevant results are three-dimensional. The most notable one
is a theorem, proved in [Hofer et al.1998], asserting that the Reeb
flow on a strictly convex hypersurface in ${\mathbb{R}}^{4}$ has either two or
infinitely many periodic orbits. In fact, more generally, this is
true for the so-called dynamically convex contact forms on $S^{3}$.
Conjecturally, this “two-or-infinitely-many” alternative should
hold for all contact forms supporting the standard contact structure
on $S^{3}$, which could be thought of as a three-dimensional analog of
Franks’ theorem; see [Hofer et al.2003] for some other related results.

The existence of infinitely many closed geodesics on $S^{2}$ also fits
perfectly into the framework of this conjecture; see [Bangert1993] and [Franks1992]
and also [Hingston1993] and [Hingston1997] and the references therein for
the original argument. Indeed, the classical Lusternik–Schnirelmann
theorem asserts the existence of at least three closed geodesics on
$S^{2}$, i.e., at least one more than is necessary from the
Floer–theoretic perspective, cf. [Katok1973] and [Ziller1983]. The geodesic flow
on $S^{2}$, interpreted as a Reeb flow on the standard contact
${\mathbb{RP}}^{3}$, should then have infinitely many simple (i.e.,
non-iterated) closed orbits or, in other words, infinitely many
geometrically distinct closed geodesics on $S^{2}$. In fact, one can
reprove the existence of infinitely many closed geodesics in exactly
this way using the variant of the Lusternik–Schnirelmann theorem
from [Grayson1989] as the starting point and then the results from
[Hryniewicz et al.2014] and [Ginzburg et al.2013] on the symplectic side of the problem;
see the latter reference for more details.

Finally, another aspect of this question is related to the so-called
perfect Reeb flows. Let us call a non-degenerate Reeb flow on a
contact manifold perfect if the differential in the contact
homology complex vanishes for some choice of the auxiliary data,
cf. [Bourgeois et al.2007]. (Thus this definition depends on the type of the
contact homology used.) For instance, a Reeb flow is perfect (for
every auxiliary data) when all closed orbits have Conley–Zehnder
index of the same parity; we refer the reader to [Gürel2015] for
numerous examples of perfect Reeb flows. One can think of
non-perfect Reeb flows as those with unnecessary periodic orbits. In
[Gürel2015] an upper bound on the number of simple periodic orbits
of perfect Reeb flows is established for many contact manifolds
under some (minor) additional assumptions. For $S^{2n-1}$, as
expected, the upper bound is $n$. However, in general, it is not
even known if a perfect Reeb flow on the standard contact
$S^{2n-1}$, $2n-1\geq 5$, must have finitely many simple periodic
orbits or, if it does, whether this number is independent of the
flow. [For $S^{3}$, this is proved in [Bourgeois et al.2007] and reproved in
[Gürel2015].]

For symplectomorphisms, the problem of the existence of infinitely
many periodic orbits breaks down into several phenomena in the same
way as for Reeb flows, although even less is known. Namely, as in
Sect. 4.1, we can, roughly speaking, single out
three types of behavior of symplectomorphisms. First of all, some
manifolds (such as ${\mathbb{CP}}^{n}$ or tori or their products) admit
symplectomorphisms with finitely many periodic orbits or even, in
some instances (e.g., ${\mathbb{T}}^{2n}$), without periodic orbits. Here a
non-obvious fact is that a surface $\Sigma_{g}$ of genus $g\geq 1$
admits a symplectic (autonomous) flow with exactly $|2-2g|$ fixed
points and no other periodic orbits; see, e.g., Katok and Hasselblatt ([Katok and Hasselblatt1995], Chap. 14)
and, in particular, Exercise 14.6.1 and the hint therein.

Then there are symplectomorphisms $\varphi$ such that the rank of
the Floer homology $\operatorname{HF}_{*}(\varphi^{k})$ over a suitable Novikov ring
$\Lambda$ grows with the order of iteration $k$. The Floer homology
groups of symplectomorphisms have been studied for close to two
decades starting with [Dostoglou and Salamon1994] and [LÃª and Ono1995], and the literature on the
subject is quite extensive (particularly so for symplectomorphisms
of surfaces); we refer the reader to, e.g., [Cotton-Clay2010] and [Fel’shtyn2012] and
references therein for recent results focusing specifically on the
growth of the Floer homology. Let us assume here, for the sake of
simplicity, that $M$ is symplectically aspherical or monotone and
that the Floer homology $\operatorname{HF}_{*}(\varphi^{k})$ is defined. Similarly to
the results in [Gromoll and Meyer1969], [Hryniewicz and Macarini2015] and [McLean2012], we have

Proposition 6.6.

Let $\varphi\colon M\to M$ be a symplectomorphism of a closed
symplectic manifold $M$ such that the sequence $\operatorname{rk}_{\Lambda}\operatorname{HF}_{*}(\varphi^{k})$ is unbounded. Then $\varphi$ has infinitely many
simple periodic orbits. Moreover, every sufficiently large prime
occurs as a simple period when the number of fixed points of
$\varphi$ is finite and $\operatorname{rk}_{\Lambda}\operatorname{HF}_{*}(\varphi^{k})\to\infty$.

Proof.

The proposition is obvious and well known when $\varphi$ is strongly
non-degenerate. (See [Cotton-Clay2010]; [Fel’shtyn2012] for more specific and stronger
results.) The degenerate case follows from the fact that the
dimension of the local Floer homology of an isolated periodic orbit
remains bounded as a function of the order of iteration, as a
consequence of Theorem3.2.
$\square$

Example 6.7.

Let $\Sigma$ be a closed surface and $\psi\colon\Sigma\to\Sigma$ be
a symplectomorphism such that $\operatorname{rk}_{\Lambda}\operatorname{HF}_{*}(\psi^{k})\to\infty$.
This is the case, for instance, when the Lefschetz number
$L(\psi^{k})$ grows; such symplectomorphisms exist in abundance.
[Proposition6.6 applies to $\psi$, but in this case a
simpler argument is available: when $L(\psi^{k})\to\infty$ the
assertion immediately follows from the Shub–Sullivan theorem,
([Shub and Sullivan1974]).] Let $P$ be a symplectically aspherical manifold with
$\chi(P)=0$, such as $P={\mathbb{T}}^{2n}$, and $\varphi\colon P\times\Sigma\to P\times\Sigma$ be Hamiltonian isotopic to $({id},\psi)$.
Then $\operatorname{rk}_{\Lambda}\operatorname{HF}_{*}(\varphi^{k})\to\infty$ and, by the
proposition, $\varphi$ has infinitely many simple periodic orbits.
However, $L(\varphi^{k})=0$, and, moreover, a symplectomorphism in the
smooth or symplectic isotopy class of $({id},\psi)$ need not have
periodic orbits at all when, e.g., $P={\mathbb{T}}^{2n}$.

Thirdly, there are symplectomorphisms with infinitely many simple
periodic orbits, but no homological growth. Here, of course, we have
the Hamiltonian Conley conjecture as a source of examples. The
authors tend to think that there should be other classes of
symplectomorphisms of this type, but no results to this account have
so far been proved. One class of symplectomorphisms which might be
worthwhile to examine is that of symplectomorphisms of
$\Sigma_{g}\times P$ symplectically isotopic to ${id}$ and with flux
vanishing on $\operatorname{H}_{1}(P)$, where $\Sigma_{g}$ is a surface of genus
$g\geq 2$ and $P$ is symplectically aspherical and not a point.

Finally, one can expect the presence of an unnecessary fixed or
periodic point to force a symplectomorphism to have infinitely many
simple periodic orbits. However, now the situation is more subtle,
less is known, and there is a counterexample to this general
principle. A prototypical (and simple) result of this type is that a
non-degenerate symplectomorphism of ${\mathbb{T}}^{2}$ symplectically isotopic
to ${id}$ has infinitely many simple periodic orbits, provided that
it has one fixed or periodic point; see Ginzburg and Gürel ([Ginzburg and Gürel2009c], Thm. 1.7). In other words, we have the following “zero or
infinitely many” alternative: a non-degenerate symplectomorphism of
${\mathbb{T}}^{2}$ isotopic to ${id}$ has either no periodic orbits or infinitely
many periodic orbits. It is interesting, however, that the
non-degeneracy condition cannot entirely be omitted, although it can
probably be relaxed. Namely, it is easy to construct a symplectic
vector field on ${\mathbb{T}}^{2}$ with exactly one (homologically trivial) zero
and no periodic points; see Ginzburg and Gürel ([Ginzburg and Gürel2009c], Example 1.10). (No
similar results or counterexamples for tori ${\mathbb{T}}^{2n}$, $2n\geq 4$,
are known.) There are also analogs of Theorem6.2
for symplectomorphisms, [Batoréo2015a]; [Batoréo2015b], applicable to manifolds
such as ${\mathbb{CP}}^{n}\times P^{2m}$, where $P$ is symplectically aspherical
and $m\leq n$.

Note in conclusion that when discussing symplectomorphisms in the
homological framework, it would make sense to ask the question of
the existence of infinitely many periodic orbits while fixing the
class of symplectomorphisms Hamiltonian isotopic to each other. The
reason is that Floer homology is very sensitive to symplectic
isotopy but is invariant under Hamiltonian isotopy. Above, however,
we have not strictly adhered to this point of view and mainly
focused on the properties of the ambient manifolds. As just one
implication of that viewpoint and to emphasize the difference with
the Hamiltonian setting, let us point that one may expect the
$C^{\infty}$-generic (or even $C^{k}$-generic for a large $k$) existence
of infinitely many periodic orbits to break down for
symplectomorphisms with a fixed flux; cf. [Herman1991a]; [Herman1991b].

Remark 6.8.

In this survey, we have just briefly touched upon the question of
the existence of infinitely many periodic orbits for Hamiltonian
diffeomorphisms and symplectomorphisms of open manifolds and
manifolds with boundary. (In this case, one, of course, has to
impose some restrictions on the behavior of the map near infinity or
on the boundary.) Such symplectomorphisms naturally arise in
applications and in physics. For instance, the billiard maps and the
time-one maps describing the motion of a particle in a
time-dependent conservative force field and/or exact magnetic field
are in this class. We are not aware of any new phenomena happening
in this setting, and our general discussion readily translates to
such maps. For instance, Hamiltonian diffeomorphisms of open
manifolds can exhibit the same three types of behavior as
symplectomorphisms of closed manifolds or Reeb flows. (After all, a
geodesic flow is a Hamiltonian flow on the cotangent bundle.) To the
best of our knowledge, there are relatively few results of
symplectic topological nature concerning this class of maps; see
Sect. 2.1 for some relevant references.

[Katok and Hasselblatt1995] Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995)